Although people living in Eretz Yisrael usually only keep a one day holiday, on Rosh Hashanah everyone keeps two days, even those living in Eretz Yisrael. Furthermoreת these two days are considered like one long day and thus carry a number of Halachic stringencies that are not applicable by other Holidays. Admur is one of the few Poskim that properly delved into this subject explaining every side of it in a very clear and logical method. Below is a free translation [with some changes to allow flow of the text] of Admur 600/1-4. A summary of Admur is included below.
1. Understanding why two days is kept even in Eretz Yisrael:
Reason why in the Diaspora they keep two days by all Holidays: The reason why in the Diaspora they always keep two-day Holidays is because in previous times the new month was sanctified based on witnesses who saw the new moon. The new month was hence not dependent on any calculation but purely on the testimony of the sighting of the moon. [The first day of the Hebrew month would be announced by the Sanhedrin based on the witnesses or lack thereof. If witnesses came forward on the 30th day then the 30th day acted as the first day of the new month. If, however, witnesses did not come forward on the 30th day then the 31st day was sanctified as the 1st of the new month. Thus, some months lasted 29 days while others lasted 30 days and it was placed in the hands of the court to announce if the 30th day or the 31st day is the 1st of the next month. Now, due to the lack of technology, in most lands out of Israel they would only hear of this announcement much later on into the next month when a messenger was able to arrive.] Thus, the communities in the Diaspora did not have knowledge as to what day the new month was sanctified in Jerusalem [and they were always in doubt as to the current date]. Due to this uncertainty of the current date, when a Holiday would approach, they would keep a two day Holiday [in order to be certain that they kept the Holiday on the correct date]. Now, although today the date of each month is set in a calendar based on calculation [and not witnesses] and hence everyone knows on which day the new month began and hence when the coming festival falls out, nevertheless in the Diaspora they keep the original custom of their ancestors which was to always keep two days.
Reason why in Israel they keep only one day: [Even in previous times] the inhabitants of Israel which lived in proximity to Jerusalem would only keep a one day Holiday, as the Beis Din in Jerusalem would send out messengers in Nissan (Sivan-See Kuntrus Hashulchan) and Tishrei to notify them as to the day that the new month was sanctified by the Sanhedrin. Thus, all the communities of close enough proximity for the messengers to reach before Yom Tov celebrated only one day. Therefore, even today the descendants [of these communities] are not required to celebrate two days as we are expert in the calculation of the new moon and know what date is fit to set the new month and count the month from it.
Reason why on R”H even in Israel they keep two days: On Rosh Hashanah even those living in Eretz Yisrael keep two days as the entire reason that by other Holidays they keep one day is because even in previous times they only kept one day and hence today we also do so as we are expert in the calculation of the new moon and know what date is fit to set the new month and count the Holidays from it. This however only applied by the other Holidays. However, regarding Rosh Hashanah even in previous times, when the month was sanctified based on testimony, they would keep two days in Eretz Yisrael, and they are hence obligated to follow the custom of their ancestors [and keep two days of R”H even today]. The reason that even in previous times they kept two days is because the month was sanctified not based on calculation but based on the establishment of the Beis Din in Jerusalem which was founded on the testimony [of the witnesses] that saw the new moon. Now, although the Beis Din would send messengers to notify everyone, nevertheless on R”H, which is Yom Tov, the messengers could not exit the Techum [outside Jerusalem, and hence only those inhabitants within the Techum would receive the news]. [Thus, even in Eretz Yisrael, in the area outside the Techum of Jerusalem, there was doubt as to when the first of Tishrei-R”H-was established, and they hence had to keep two days due to doubt.]
When did they begin Yom Tov Sheiyni of Rosh Hashanah in Eretz Yisrael?
Some write that the two-day celebration even in Israel only began in the times of Rebbe Yochanon ben Zakaiy. Prior to this time, it was only celebrated as one day [in years applicable, as explained above]. Others however right at the two-day celebration in Israel began in the times of Ezra.
2. The two days of R”H are considered one long day:
Difference between the two days of R”H and the two days of other Holidays: In truth even in the Diaspora the two days of R”H are different than the two days of other Holidays. By the other Holidays even the ancestors only kept two days in the Diaspora due to doubt, as they did not know the date that the month was set in Eretz Yisrael. Hence only one day is truly the holiday and if the 1st day is Yom Tov then the 2nd day is a regular weekday, if the 2nd day is Yom Tov the first day is a regular weekday. However the two days of R”H are not so and they are considered like one long day and they are one period of Holiness even in Eretz Yisrael.
The ramifications: The practical ramification of this matter is regarding Nolad or Muktzah that occurred on the first day of Yom Tov if it may be benefited from on the second day. By other Holidays if Nolad or Muktzah occurred on the first day of Yom Tov it is permitted on the 2nd day. However on R”H if Nolad or Muktzah occurred on the first day it is also forbidden on the second day being that they are considered like one long day. Another ramification is regarding the laws of medical treatment. Another ramification is regarding the laws of a food that was brought from outside the Techum on the first day of Yom Tov. Another ramification is regarding one who forgot to do Eruv Tavshilin on Erev Yom Tov whether he may do so on the first day of Yom Tov through making a Tnaiy. Another ramification is regarding whether one is to say Shehechiyanu on the second day of Yom Tov by candle lighting and Kiddush.
Being stringent at all times: The two days of R”H are only considered like one day regarding being stringent in the ramifications explained above, and not regarding being lenient in those matters that seemingly we should be lenient on if they are considered one day. Thus, one may not prepare on the first day of R”H on behalf of the second day of R”H, just as is the law by any Holiday.
The reason it is considered one long day
Even in Jerusalem they sometimes kept two days: The reason that on R”H the two days are considered like one, despite the fact that we are knowledgeable of the date of the month and we hence know that the first day is the main day, is because even the Beis Din themselves which would sanctify the month based on sighting the new moon would at times keep two days. How did this occur? At first when the 30th of Elul arrived the entire Jewish people and the Beis Din would keep Yom Tov and not do Melacha starting from the night [of the 30th] as perhaps the witnesses would come the next day and testify that they saw the moon that night hence causing the 30th to be Rosh Chodesh Tishrei which is Rosh Hashanah, and hence consequently the night before was Yom Tov. The Beis Din would sit during the day of the 30th and wait for witnesses which saw the moon to come forward. Whatever hour the witnesses would arrive throughout the day, until night, their testimony would be accepted, and the month was sanctified that day. That day would be Rosh Hashanah and from that day and onwards they would count for Yom Kippur and Sukkos. Usually the witnesses would arrive before the time of Mincha which is the hour that the afternoon Tamid was offered. One time the witnesses delayed coming until after Mincha and the Tamid sacrifice was offered and the Levites sang the song of the wrong day, singing the song of the weekday [as opposed to the song of Musaf R”H] as they thought that witnesses would no longer arrive that day, but when they did arrive it caused the day to retroactively be Holy. From that day and onwards the Sages established that the Beis Din would no longer accept testimony past Mincha and when witnesses would arrive after Mincha they would make that day part of the previous month, and it would be the 30th of Elul, and the next day [would be the first of Tishrei-R”H] from which they would count to Yom Kippur and Sukkos. Nevertheless, despite [that the 30th day is being counted as the last day of Elul and hence] the Holidays are not being counted from then and it is thus a weekday and Yom Tov will only begin that night, nevertheless due to a Rabbinical decree they would not belittle that day and they would guard the remainder of the day, until night, following all the prohibitions of the Holiness of Yom Tov. Thus while the 30th day was the Rabbinical Yom Tov the next day was the Biblical Yom Tov and R”H. These two days are considered one state of holiness and like one long day and are both called the Yom Tov of R”H, the first day being Rabbinical while the second day being Biblical.
[The above explanation however raises the following question: Although it may be true that there were years that even in Jerusalem they kept two days, the first as Rabbinical and the second as Biblical, nevertheless there were many years that in Jerusalem they did not keep two days as the witnesses arrived before Mincha and hence the month was sanctified on the 30th and the next day was already the 2nd of Tishrei. Hence why should the two days of R”H of the entire world now be considered one long day simply because at times even in Jerusalem it was all one long day. It is this question that the next part of Admur addresses]
The affect on the two days kept outside of Jerusalem: The areas that were outside of the Techum of Jerusalem [and hence the messengers could not arrive on R”H] would need to celebrate two days of R”H every year [due to doubt as explained above in A]. Now [this doubt also included that] perhaps the witnesses would arrive after Mincha and hence both days are holy and considered like one long day, as they both contain the same holiness of R”H. It was only after Yom Tov that the messengers would leave and tell all the inhabitants of Eretz Yisrael the day that the month was established, and everyone would count from the correct date for Yom Kippur and Sukkos. [Thus, in conclusion all the inhabitants that were outside of the Techum of Jerusalem had to keep two days of R”H as if it were one long day due to the doubt of whether witnesses arrived after Mincha.]
The resultant law of today: The above system was always followed until the nullification of the Sanhedrin and the sanctifying of the moon based on testimony became abolished, thus forcing them to sanctify the months based on calculations. From that point and on everyone in the world counts the first day of R”H as the first of Tishrei and from that day they count the days for Yom Kippur and Sukkos. The reason for this is because based on calculation the month of Elul is always 29 days [and hence the 30th is always the first of Tishrei]. Now since we need to follow the custom of our ancestors that was followed during the times that they sanctified the new month based on testimony and hence they would celebrate two days of R”H even in Eretz Yisrael, and they would consider it one long day with one set of holiness, therefore we also must follow this and consider the second day as part of one long day in order not to belittle the second day and be lenient on it regarding matters that our forefathers were not lenient. Thus just like our forefathers had to prohibit Nolad on the second day of R”H due to the doubt that perhaps the witnesses arrived after Mincha and it is hence one long day, similarly we must prohibit Nolad on the 2nd day even though that the reason is no longer applicable today being that we set the month based on calculation and the first day of R”H is always Biblically the main day of Yom Tov.
Why even in Eretz Yisrael they keep two days: Although people living in Eretz Yisrael usually only keep a one-day holiday, on Rosh Hashanah all people including those living in Eretz Yisrael keep two days. The reason for this is as follows: The reason why in lands out of Israel two days are kept by all Holidays is because in the times of the Temple, the first day of the Hebrew month would be announced by the Sanhedrin based on the appearance of the moon. Some months lasted 29 days while others lasted 30 days and it was placed in the hands of the court to announce if the 30th day or the 31st day is the 1st of the next month. Now, being that news did not travel around very quickly, in most lands out of Israel they would only hear of this announcement much later on into the next month. Thus, they were in constant doubt in the correct date of the month. Because of this when a Holiday would approach, they would all keep a two day Holiday in order to be certain that they kept the Holiday on the correct date. In Israel however the announcement of the Sanhedrin was able to be spread and thus they had no doubt as to the correct date of the month and thus had no need to keep two days. However, regarding Rosh Hashanah being that it is on the 1st of the month, only those living within the Techum in Jerusalem, by the quarters of the Sanhedrin, would hear of the announcement. Thus, even in Israel all people living outside Jerusalem would keep a two day holiday out of doubt to which day has been announced the 1st of the month.
Why it is one long day: Even in Jerusalem many times two days were kept being that the announcement could take place throughout the entire day of the would be 1st of the month, and out of doubt people would have to keep that first day as a holiday. If no announcement was made and only the next day was declared the 1st of the month then in total they would end up keeping two days. Furthermore, at times the two days were not only kept out of doubt but based on a Rabbinical decree, as if witnesses of the new moon would arrive to the Sanhedrin on the 30th day after Mincha, then their testimony would not be accepted, and the 1st of the month would only be the next day. However, the Sages decreed that since they arrived on the 30th day therefore also the 30th day would be considered a holiday, and thus both days had the status of Rosh Hashanah, the first day Rabinically while the second day Biblically. It is for this reason that the two days of Rosh Hashanah are considered one long Holiday. This has Halachic ramifications regarding the laws of Nolad and Muktzah that occurred on the first day of Yom Tov in which case due to above, the prohibition applies even on the second day.
Sparks of Kabala:
The reason that today even in Eretz Yisrael we always keep two days of R”H is because after the destruction of the Temple we are unable to affect all the elevations of the Penimiyus and Chitzoniyus of Malchus in one day. However, prior to the Churban we were able to elevate everything in one day in Eretz Yisrael and hence they only kept two days when the witnesses arrived after Mincha.
3. Days on which Rosh Chodesh will never fall:
The Sages established that Rosh Hashanah, which is Rosh Chodesh Tishrei, cannot fall out on certain days of the week even if the Molad is on that day. It cannot fall on Sunday, Wednesday or Friday. In the event that the Molad falls on one of these days, Rosh Chodesh is pushed off a further day. Some Rishonim explain the reason for the avoidance of the above weekdays is because doing so would cause Yom Kippur to fall a day before or a day after Shabbos and they did not want there to be two consecutive days of prohibited food preparation. Likewise, they did not want Hoshana Rabbah to fall on Shabbos which would nullify the Mitzvah of Arava. The Rambam however explains the reason this is done is in order to have Rosh Chodesh fall on the Cheshbon Haamiti of when the moon will be seen and not on the Molad, which is the Cheshbon Haemtzai. This decree effected that the Rosh Chodesh of every month cannot fall on certain days, as once we force Rosh Chodesh Tishrei to only fall on certain days, and the remaining months follow an automatic pattern of Malei and Chaseir, it effects that every Rosh Chodesh has only certain days on which it can fall. Practically, the following are the dates in which Rosh Chodesh can and cannot fall on:
 Admur 600:1; Michaber 601:2
 600:1; See also Levush 600; Beis Yosef 600; M”A 601:1; Rambam Hilchos Kiddush Hachodesh 5:4-12; Kaf Hachaim 601:5
 Vetzaruch Iyun Gadol as a) Even in Jerusalem they would have to keep two days in all years that witnesses did not come due to doubt that perhaps witnesses would arrive that day, and in the end if the witnesses did not arrive then the next day would be R”H. Thus, why are we saying the two days were kept only outside of the Techum? Perhaps however one can answer the above by saying that since there were years that in Jerusalem they kept only one day due to witnesses arriving and hence the next day was a weekday, the 2nd of Tishrei, and since the two days were not kept every year it cannot serve as a reason to keep two days everywhere in Eretz Yisrael, and hence Admur does not mention it. However, this leads to the second question: B) If witnesses did not arrive then it is understood that everyone, including the inhabitants of Jerusalem, kept two days due to the doubt. If witnesses did arrive then in Jerusalem, they kept one day and the next day was a weekday. If so, then why didn’t they send messengers immediately after the first day ended when there is no longer a prohibition of leaving the Techum as they know that it is a weekday. Thus there is no reason why the people outside the Techum of Jerusalem should keep two days of Yom Tov due to doubt in those years that witnesses arrived on the first day, as the messengers could easily arrive on their second day and tell them that it is no longer Yom Tov? Perhaps one can answer this question as follows: a) There were no cities close enough to Jerusalem that the messengers could arrive before the end of the second day if they did not start traveling already on the first day past the Techum. B) Alternatively since all the places outside the Techum had to at least begin the 2nd day following all the prohibitions of Yom Tov due to doubt, and many years they had to keep the entire second day due to lack of witnesses arriving, therefore also today the inhabitants of Jerusalem always keep two days. Vetzaruch Iyun!
 See Menachem Meishiv Nafshi Vol. 1 p. 85
 Hagahos Rav Chaim Vital
 Tosafus Rosh Hashanah 19b
 600:3; Tur; Levush; Taz 600:1
 An egg laid on Yom Tov: By other Holidays an egg that was laid on the first day of Yom Tov is permitted on the 2nd day, as if today is holy then tomorrow is a weekday and it does not contain the prohibition of Muktzah and Nolad. If tomorrow is holy, then today is a weekday and the egg was hence prepared from Erev Yom Tov. Therefore even the descendants of the inhabitants of the Diaspora which know the date of the month and know that the first day is the main day of Yom Tov while the 2nd day is a Rabbinical Holiday [and is no longer a doubt] nevertheless this egg does not have the same status as an egg that was laid by two Holidays that proximate each other, such as Shabbos and Yom Tov, in which case if it was laid one day it is forbidden on the next. [see 513:8-9] The reason is because since the entire reason of them keeping two days is due to their ancestor’s custom therefore, they are not required to be more stringent that their ancestors to which this egg was permitted on the 2nd day as explained. [600:2] However on R”H if an egg is laid on the first day, or a food was detached from the ground on the first day [see 515:9], or an animal was trapped on the first day [see 497:9; 515:6], then just like it is forbidden due to Muktzah and Nolad on the first day so too it is forbidden on the second day being that they are considered like one long day. This however is with exception to burying a corpse on the 2nd day as is explained in 526:6. [600:3; Michaber 600:1; 479:9 [trapped animal]; 513:8,10,14 [egg laid]; 515:9 [food detached from ground]
On the first day of a regular Yom Tov these laws follow the same laws as Shabbos [see our Sefer The laws of Shabbos Vol. 2 “Medicine and First Aid” chapter 2] that even if one is bedridden a treatment involving Rabbinical prohibition may only be done with a Shinui. However, on the second day of Yom Tov we are lenient to allow even a Jew to perform Rabbinical prohibitions of medical treatment without a Shinui even if the person is not bedridden but is in a lot of pain. However, on R”H the second day follows the same laws as the first day for all matters. [ibid]
 515:1 that according to all opinions on R”H one may not benefit until after the second day, even in a case of guests. On other Yomim Tovim however it is disputed if one may benefit on the second day and we are lenient in a case of need as explained there.
 527:22; By a regular Yom Tov that falls on Thursday, Friday one may make an Eruv Tavshilin on Thursday using a Tnaiy that “if today is Yom Tov tomorrow is a weekday etc.”. [527:21] However on R”H one cannot make an Eruv Tavshilin being that it is all considered one long day. [527:22]
 601:5; On regular Holiday one says Shehechiyanu on both the first and second day. [ibid; 490:1] However on the second day of R”H since it is considered one long day it is disputed whether Shehechiyanu is to be recited. See Chapter 1 Halacha 24 for the full discussion of this matter.
 Seemingly this refers to Mincha Ketana which was at the 9.5 hours into the day. [See M”A 233:2]
 See Rambam Hilchos Temidim Umusafim 6:9; Tamid 33b
 Vetzaruch Iyun regarding whether the inhabitants of Jerusalem did Melacha starting from Mincha and onwards if they did not hear that witnesses arrived, and the Beis Din sanctified the month. From the fact Admur states that only the song of the Kohanim was confused implies that everyone refrained from Melacha due to doubt, otherwise there would be a much greater reason to prevent accepting testimony after Mincha, as this would cause people to desecrate Yom Tov retroactively.
 The reason the Beis Din has power to uproot the Mitzvah of sanctifying the day based on testimony: The Beis Din has power to do the above as the verse states “Aleh Moadeiy Hashem Asher Tikreu Osam” The word Osam is written without a “vav” and is hence read “Atem”. This comes to teach us that even if the Beis Din purposely calls it on the wrong day it is valid. [ibid]
 The reason: Since from the letter of the law the 30th day was Biblically supposed to be the Holiday [first of Tishrei] and it is only due to that Beis Din has permission to delay the sanctifying of the month that it was delayed therefore they decreed to keep also the 30th as Yom Tov. [ibid]
 Shaar Hakavanos 91; Kaf Hachaim 601:5; See Admur Basra 1:8
 Michaber 428:1-3; Rambam Kiddush Hachodesh 7
 Rambam ibid 7:1
 Raavad 7:7
 Rambam ibid 7:7