Throughout the days between R”H and Yom Kippur one should increase in prayer and supplication.
The widespread custom is to recite Selichos every day between R”H and Yom Kippur, with exception to Shabbos. [However the Chabad custom is not to say Selichos during Aseres Yimei Teshuvah with exception to Tzom Gedalia in which the typical fast day Selichos is recited.]
The confession prayer in Selichos: [Those that recite Selichos during Aseres Yimei Teshuvah] say the confession prayer [i.e. Viduiy] three times every day prior to dawn. This is with exception to Shabbos in which case it is completely omitted, and Erev Yom Kippur in which case it is said only one time. [Practically the Chabad custom is to only recite the confession prayer of Ashamnu one time whenever Selichos is recited. Furthermore, as stated above, it is not the Chabad custom to recite Selichos at all during Aseres Yimei Teshuvah.]
C. Shir Hamaalos
One recites the psalm of Shir Hamaalos Mimamakim between Yishtabach and Barchu throughout the ten days of repentance. See the section on Rosh Hashanah Chapter 1 Halacha 3 for the full details of this subject!
D. Shemoneh Esrei:
In Shemoneh Esrei one must remember to add the following statements: Zachreinu; Mi Kamocha; Hamelech Hakadosh; Hamelech Hamishpat; Ukesov; Ubesefer; Oseh Hashalom. For further details on the laws relating to Shemoneh Esrei see the section on Rosh Hashanah Chapter 1 Halacha 5-6!
E. Changes in Kaddish:
At the conclusion of Kaddish one says “Oseh Hashalom” in place of “Oseh Shalom”. See the section on Rosh Hashanah Chapter 1 Halacha 5D!
F. Avinu Malkeinu:
It is accustomed to recite Avinu Malkeinu [each day after the Shemoneh Esrei] in the morning [by Shacharis] and in the evening [by Mincha]. [It is recited even if one is not Davening with a Minyan. The custom is to open the ark prior to its recital. The custom is to stand during its recital. The Chabad custom is to recite it in middle of Tachanun, prior to the words Veanachnu Lo Neida. One continues straight to Veanchnu Lo Neida after Avinu Malkeinu and omits the short Avinu Malkeinu prayer normally recited prior to Veanachnu Lo Neida. Others however are accustomed to recite it at the end of Tachanun, prior to Kaddish.]
G. Bris or Chasan:
If a Bris is taking place during the days between R”H and Yom Kippur [or a Chasan is present] then Tachanun [Lamnatzeiach and Keil Erech Apayim] is omitted [in that Shul] although Avinu Malkeinu is recited. [Although there were years that the Rebbe omitted Avinu Malkeinu during Aseres Yimei Teshuvah if a Chasan was present, in other years he recited it.]
Is one to hit his chest upon saying “Chatanu Lefanecha?
Is Avinu Malkeinu recited in the house of an Avel?
Thirteen attributes of mercy by Kerias Hatorah:
It is not the Chabad custom to recite the thirteen attributes of mercy throughout the ten days of repentance when the Ark is opened for Kerias Hatorah. It is only recited on Yom Tov. Others have a custom to do so each time the ark is opened during these ten days.
 602/2; Michaber 602/1
 Tur 582 in name of Rav Haiy Gaon; Rama ibid; Admur ibid regarding the three confession prayers recited in Selichos; Admur 604/4 regarding Erev Yom Kippur “We are not accustomed to say such lengthy Selichos in contrast to the other days”; See also Siddur Admur regarding Kaparos of Erev Yom Kippur “by Ashmuros Haboker after Selichos”; See Rebbe in Shaar Hamoadim Elul 75
 The Tur 602 writes that in Sefaradi the custom was to say Selichos even on Shabbos, however the Beis Yosef writes there that this is no longer accustomed. [Kaf Hachaim 602/3]
 Sefer Haminhagim p. 122 [English]; Igros Kodesh 2/252; Shulchan Menachem 3/113; See Otzer Minhagei Chabad p. 161 that the Rebbetzins did say Selichos during these ten days, and that in Lubavitch there were Chassidim that recited Selichos although majority did not. Likewise the Rabbeim, starting at least from the Tzemach Tzedek, did not recite Selichos.
The reason: The reason behind this custom is because Aseres Yimei Teshuvah is a time for doing the action of Teshuvah, while the speech of Teshuvah pertained only up until Rosh Hashanah. [Rebbe Rayatz in the name of the Tzemach Tzedek Sichos 1948; brought by Rebbe in Sefer Hasichos 5750 vol. 1 p. 357 footnote 132]
 602/3; Rama 602/1; Rokeiach; Levush 581; Darkei Chaim Veshalom 703
 The reason it is not recited on Yom Kippur more than one time is because one will recite the Viduiy by Mincha and by Maariv [for a total of three times]. [Rokeiach 214; Gr”a] Alternatively the reason is because Erev Yom Kippur is a day of Yom Tov. [Gr”a]
Other Opinions: The Beis Yosef 604 writes that the custom is to say Viduiy three times even on Erev Yom Kippur. Practically we do not rule like this opinion. [Kaf Hachaim 602/14]
 Sefer Haminhagim p. 115 [English] based on Arizal in Peri Eitz Chaim 51; Peri Eitz Chaim 5/8; So rules also Ashel Avraham Butchach; Gr”a in Maaseh Rav; Kaf Hachaim 602/13; See Otzer Minhagei Chabad 66
The reason: As one who says Viduiy more than one time in a row is like an animal that is returning to sin. [Arizal ibid, brought in Kaf Hachaim 602/13]
 602/2; Levush 602; This is originally an Ashkenazi custom but is now accustomed even amongst Sefardim. [Kaf Hachaim 602/2]
Background of Avinu Malkeinu:
The Gemara in Tanis 25b states that one time a fast was decreed and they were not answered until Rebbe Akiva Davened for the Amud and recited the prayer of Avinu Malkeinu [the stanza of “Ein Lanu Melech Ela Ata” and of “Lemancha Rachem Aleinu”], upon which he was immediately answered. After the affect of this prayer was witnessed more stanzas were added and it was established to recite it during Aseres Yimei Teshuvah. [Beis Yosef in name of Kol Bo; Kaf Hachaim 584/3; See Mateh Moshe 801; Shaar Hakolel 42/12] Some say the reason we recite Avinu Malkeinu is in correspondence to the middle blessings of Shemoneh Esrei that are omitted on Rosh Hashanah. [Levush brought in Kaf Hachaim 584/7]
Repeating the middle stanzas after the Chazan: It is not our custom to repeat the middle stanzas aloud after the Chazan. [See Shaar Yissachar p. 219 which protests against those which do so being that they only emphasize the stanzas dealing with physical benefit while those dealing with spiritual matters are said in regular tone.] However there are Poskim which write that the custom is to say it aloud. [Mateh Efraim 602/13]
 Tanya 77; Kaf Hachaim 602/5
 Kitzur Shlah; Shvus Yaakov 3/42; Elya Raba 584/5; Shaareiy Teshuvah 584/2; M”E 584/13; Kaf Hachaim 584/11; Daas Torah 584/1
Other Opinions: Those that are not accustomed to recite Avinu Malkeinu unless they are with a Minyan have upon what to rely as the source of Avinu Malkeinu is from Rebbe Akiva who said it with a Minyan. [Elya Raba ibid brought in Kaf Hachaim ibid; Daas Torah 584/1]
 Mateh Efraim 619/46; Otzer Minhagei Chabad 199 in name of Sefer Haminhagim
 Makor Chaim of Chavos Yair 602; Piskeiy Teshuvos 602/2
The reason: As it was established to be recited corresponding to Shemoneh Esrei. [ibid]
 Otzer Minhagei Chabad 6
 602/2; 584/5; Rama 584/1; 602/1; Rivash 512; Kneses Hagedola 584/2; Peri Chadash; Kisei Eliyahu 584/3
Other opinions: Many Poskim rule that Avinu Malkeinu is to be recited even on Shabbos. [Rashbatz 3/186 brought in Beis Yosef; Hatanya; Mateh Yehuda 584; implication of Arizal in Shaar Hakavanos; see Kaf Hachaim 584/8] Based on the Arizal ibid it appears that one is to recite all the stanzas of Avinu Malkeinu, with exception to the one’s that mention sin, even on Shabbos Shuva, and so is the custom of the Beis Keil community in Jerusalem. [Kaf Hachaim 582/16] The Kaf Hachaim 584/8 concludes: It seems that the Ashkenazi custom is to omit it while the Sefardi custom is to say it and for this reason the Michaber omitted this ruling from his Shulchan Aruch. Each community is to follow their custom.
 The reason: As it is forbidden request one’s needs on Shabbos. [ibid; Ran] Some however say the reason we recite Avinu Malkeinu in the first place on Rosh Hashanah is in correspondence to the middle blessings of Shemoneh Esrei that are omitted on a weekday Holiday, however on Shabbos since in any event we do not recite the weekday blessings in the prayer, therefore there is no need to recite Avinu Malkeinu. [Levush brought in Kaf Hachaim 584/7]
 584/5; 602/2; Meil Tzedaka; Rivash 512; M”A 584/3; M”E 584/11; M”B 584/4; Shaareiy Tziyon 584/7; P”M 584 A”A 6; Kaf Hachaim 584/7
Other opinions: From the Levush it is implied that Avinu Malkeinu is recited on Mincha Erev Shabbos. [Kaf Hachaim ibid]
 The reason: From Admur it is implied that this is for the same reason mentioned above, that it is forbidden to make personal requests on Shabbos. Vetzaruch Iyun as it is not yet Shabbos and hence should not carry this prohibition? [See Kaf Hachaim 584/7]
 602/4; Rama 602/2 in name of Minhagim
 This includes the paragraph of Vehu Rachum recited on Mondays and Thursdays. [ibid]
 Piskeiy Hassidur 151 and 179; Sichas Kodesh 6th Tishrei 1975 based on Admur in Siddur that Lamnatzeiach is never recited when Tachanun is omitted. This ruling in the Siddur is based on Kneses Hagedola 131/3; Peri Chadash 131/1; Mamar Mordechai 131/6; Kaf Hachaim 581/78 states that so is the custom of the Sefaradim.
Ruling of Admur in the SHU”A: In the Shulchan Aruch, Admur rules like Rama 131/1 that Lamnatzeiach may be recited even on a day that Tachanun is omitted. [Admur 429/12; 602/4 [in parentheses]]
 Siddur Tehillas Hashem [from 1978 and onwards with the Rebbe’s approval] based on Sichas Kodesh 6th Tishrei 1975 that the Rebbe equates Kel Erech Apayim with Lamnatzeiach. See Shulchan Menachem 3/293; Glosses of Rav Raskin p. 165; Hiskashrus 438 p. 17; To note however that the Rebbe himself did recite Kel Erech Apayim even when Tachanun was not recited. [Rav Raskin ibid]
Ruling of Admur: In the Shulchan Aruch [429/12; 602/4] Admur rules that Kel Erech Apayim is recited with exception for Erev Pesach. This follows the ruling of the Hagahos Maimanis; Darkei Moshe 602 brought in Kaf Hachaim 602/12. [Some explain that it is for this reason that Admur in 602/4 placed this ruling in parentheses, as it is not brought in previous Poskim in Shulchan Aruch. [Shulchan Menachem ibid footnote 2] Vetzaruch Iyun as it is brought by Admur in 429/12 without parentheses, and is likewise clearly implied from the Rama 429/2] To note that also in the Siddur before Kel Erech Apayim Admur lists a number of days that it is to be omitted on, thus implying it is not always omitted when Tachanun is not said.
Ruling and explanation of Divrei Nechemia: The Divrei Nichemia 131/9 brings three opinions regarding the saying of Keil Erech Apayim:1) It is only omitted by a day that Halel is recited or a day that has the status of a Yom Tov like Erev Pesach. [This is the opinion of the Rama 429/2] 2) It is omitted on every Erev Yom Tov and on Isru Chag. 3) Every day that Tachanun is omitted so is Lamnatzeiach. [so rules Peri Chadash 131] The Divrei Nechemia concludes that by Lamnatzeiach we rule like the third opinion, however by Keil Erech Apayim we rule like the opinion that rules it is to be recited.
The reason behind the Rebbe’s ruling: The Rebbe explains that the ruling in the Shulchan Aruch of Admur follows the ruling of the Rama that differentiates between the laws of Tachanun and that of Lamnatzeiach. However according to Admur in the Siddur that rules Lamnatzeiach is always omitted when Tachanun is not said then likewise Kel Erech Apayim is to be omitted in all days that Tachanun is not said. [Sichas Kodesh ibid, printed in Shulchan Menachem ibid] As for the reason why Kel Erech Apayim is omitted this is because it mentions sin, and we do not desire to mention sin on any day that Tachanun is omitted. [See Shaar Hakolel 36/2; Shulchan Menachem ibid footnote 5]
 6th Tishrei 1975 the Rebbe did not say Avinu Malkeinu because there was a Chasan in the Minyan and the Rebbe than explained it in a Sicha on that day that according to Admur in the Siddur Avinu Malkeinu is never recited when Tachanun is omitted. [See Shulchan Menachem 3/293; Otzer Minhagei Chabad p. 166; Glosses of Rav Raskin p. 165]
 Shulchan Menachem ibid footnote 7 that there were many instances after this Sicha was recited that the Rebbe said to recite Avinu Malkeinu even when a Chasan or Baal Bris was present. See also Sichos Kodesh 1980 p. 865 “On Monday Tzom Gedalia after Selichos the Chazan began saying Kaddish. The Rebbe motioned to him and told him to say Avinu Malkeinu. The Rebbe was then told that there is a Chasan.” [It does not state what they did in the end. However it seems clear that the congregation in 770 accepted this earlier ruling of the Rebbe that Avinu Malkeinu is not recited. However the Rebbe himself seems to have retracted from this ruling as he knew that a Chasan was already there as why else would the congregation omit Tachanun prior to Selichos. Hence it seems the Rebbe which certainly knew there was a valid reason not to say Tachanun ruled that nevertheless Avinu Malkeinu is to be recited.] See Hiskashrus 438 p. 17 footnote 9
 Kaf Hachaim 584/5; See M”E 584/11; P”M 584 A”A 2
 The new Tehillas Hashem [Kehos Israel] records that the Rebbe was accustomed to hit his chest upon saying these words in Avinu Malkeinu.
 M”E 602/11; Otzer Minhagei Chabad p. 160 based on directive of Rebbe; Piskeiy Teshuvos 602/2
 Otzer Minhagei Chabad 10