When:

The meat of the Pesach sacrifice on the night of the 15th.

Not before nightfall/Tzeis Hakochavim:[1] One does not fulfill the Mitzvah of eating the Karban Pesach prior to nightfall [i.e. Tzeis Hakochavim].[2] Furthermore, if one ate a Kezayis of the meat prior to nightfall, he transgresses a Biblical command.[3]

Before midnight:[4] One is required to eat it to midnight.[5] [If one eats the meat only after midnight, some Poskim[6] rule he does not fulfill his obligation and has thus lost the Mitzvah. Other Poskim[7] rule he can nevertheless fulfill the Biblical Mitzvah up until Alos Hashachar. Other Poskim[8] rule that this matter is left in question, and hence one is to be stringent like both opinions, and eat the meat without a blessing. Practically, if one did not yet eat the meat and it is already after midnight, he is to eat the meat without the blessing.[9] If it is already close to midnight and one did not yet eat the meat, he is to skip from wherever he is holding and eat the meat before midnight.]

When during the meal:[10] The Pesach sacrifice is to be eaten at the end of the meal in order so it be eaten with satiation, which means that the Pesach sacrifice should be the conclusion of all ones satiated eating in order so one receive pleasure from the eating and it be considered important to him.[11] However, it is also eaten at the beginning of the meal together with the Matzah and Maror.[12]

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[1] See Admur 472:2; Levush 472:1; Terumas Hadeshen 137; Tosafus Pesachim 99b in name of Mahariy Krubil; Rosh 10:2; Shulchan Gavoa 472:3; P”M 472 M”Z 1; Kaf Hachaim 472:7

[2] The reason: The eating of Matzah is connected [Hekesh] to the eating of the Pesach lamb, as the verse [Bamidbar 9:11] states “Al Matzos Umerorim Yochluhu,” and by the Pesach lamb it states [Shemos 12:8] “Veachlu Es Habasar Balayla Hazeh” from which we learn it must be eaten by actual night. Now, since the eating of the Pesach lamb can only be fulfilled at night so too the eating of Matzah.  [Admur 472:2; Levush 472:1; Terumas Hadeshen 137; Tosafus Pesachim 99b in name of Mahariy Krubil; Rosh 10:2]

[3] Rambam 8:5

[4] Admur 458:2 “One is required to eat the Matzah prior to midnight, just like the law regarding the Karban Pesach which may only be eaten until midnight,”; Admur 639:20 “And one is to beware to eat in the Sukkah prior to midnight, just as is required regarding eating Matzah”; 477:6 regarding Afikoman; See 472:2 regarding the Hekesh being used to obligate eating Matzah after Tzeis Hakochavim [Vetzaruch Iyun why the end time was not mentioned in this Hekesh, and why we make the Hekesh for the end time only regarding Matzah and not the four cups, also Tzaruch Iyun why in 475:20-23 no mention of Chatzos was made; See here: http:::shulchanharav.com:text.asp?index=128712&arc_id=138425]; Michaber 477:1 and Beis Yosef 477 regarding Afikoman;  Degul Merivava 477; Shaareiy Teshuvah 477:1; M”B 477:6 “One must be very careful”; Biur Halacha 477; Kaf Hachaim 475:64; 477; Piskeiy Teshuvos 477:4

[5] The reason: As just like the Karban Pesach may only be eaten until midnight, so too the Matzah may only be eaten until midnight, as the verse states Al Matzos Umerorim Yochluhu, thus teaching us that the eating of Matzah is connected, and follows similar laws, to the Pesach [Admur 458:2] This follows the opinion of Rebbe Elazar Ben Azaria, brought next.

[6] Reb Elazar Ben Azaria in Pesachim 120b; Zevachim 57b; Tosafus Zevachim ibid; Megillah 21a; Piskeiy Tosafus Pesachim 334; Zevachim 33; Rabbeinu Chananel in Arvei Pesachim; Semag in name of Riy; Mordechai end of Pesachim; Rabbeinu Yerucham 5:4; Conclusion of Or Zarua; Hapardes 9:16-26 “It is forbidden to eat Matzah after midnight”; Kol Bo 90 in name of Riy; See Biur Halacha 477:1 “Viyihei Zahir”

[7] Rebbe Akiva in Pesachim and Zevachim ibid; Rambam Karban Pesach 8:15; Chametz Umatzah 6; Rif [brought in Mishkanos Yaakov 139] Rav Hamaggid; Baal Haitur; Or Zarua in name of Rav Yaakov of Krubil; Kol Bo 90 in name of Rav Shmuel Shakli; See Biur Halacha 477:1 “Viyihei Zahir”

[8] Rosh; Rashba; Ran end of Megillah chapter 2; Rokeiach

[9] Degul Merivava 477; Shaareiy Teshuvah 477:1; P”M 477 M”Z 1; Derech Hachaim; Tov Ayin 38:87; Ikarei Hadat 19:14; Misgeres Hashulchan 119:11; Aruch Hashulchan 477:5; M”B 477:6; Biur Halacha ibid; Kaf Hachaim 473:16; 475:64; Beis Halevi 1:34; Nitei Gavriel 91:1; Piskeiy Teshuvos ibid

Other opinions: Some Poskim rule that the final ruling of the Talmud is like Rebbe Akiva that the Mitzvah of Matzah applies throughout the night, until Alos Hashachar. [Mishkanos Yaakov 139; Or Yisrael 475; Or Sameiach 6:1; Sdei Chemed Mareches Chametz Umatzah 14:11; See Nitei Gavriel ibid footnote 1] According to all, the blessing may no longer be said once Also Hashachar has arrived. [Hisorerus Teshuvah 269; Piskeiy Teshuvos ibid]

[10] Admur 477:1; Rambam 8:9; Rashbam Pesachim 119b; Pesachim 70a; Levush 4776:1

[11] The reason: The reason for this requirement is because all Kodshim are required to be eaten with great importance and aristocracy, as the verse [Bamidbar b18:8] states “Kodshei Bnei Yisrael Lecha Nasatim Lemishcha.” This means to say that the meat had to be eaten in a way of greatness and aristocracy like kings and princes eat it. [Admur ibid; See Sotah 15a]

Other reasons: Some suggest that the reason that the Karban Pesach was eaten at the end of the meal is because it is forbidden to eat anything afterwards, as well as because if it were eaten when people were still hungry, they may come to break bones from the offering. [Levush 477; Kaf Hachaim 476:3]

[12] Rambam Chameitz Umatzah 8:6