Gemara

10a (R. Yochanan citing R. Shimon ben Yehotzadak): On 18 days every year, individuals say full Hallel: all eight days of Sukos, the eight days of Chanukah, the first day of Pesach, and Shavu'os.

10b: "Ha'Shir Yihyeh Lachem k'Leil Hiskadesh Chag" teaches that song (full Hallel) is required only on days when (even) the night is Mekudash (i.e. Melachah is forbidden).

11a: Chachamim use "Ha'Leviyim... u'Lechaper Al Bnei Yisrael" to teach R. Elazar's law;

(R. Elazar): Just like Kaparah must be during the day, also Shirah.

Pesachim 95b (Mishnah): One must sing Hallel while eating Pesach Rishon (but not while eating Pesach Sheni).

117b (Mishnah): We say Birkas ha'Mazon over the third cup;

Inference (Rav Chanan): Birkas ha'Mazon must be said over a cup of wine. (If not, three cups would suffice!)

Rejection (Rava): No. Chachamim enacted to drink four cups of wine like free men, and they associated each of them with a Mitzvah.

(Mishnah): We finish Hallel over the fourth cup, and say Birkas ha'Shir.

118a (Rav Yehudah): Birkas ha'Shir is 'Yehalelucha Hash-m Elokecha' (the closing Berachah of Hallel);

(R. Yochanan): 'Nishmas Kol Chai' is also Birkas ha'Shir.

Ta'anis 28b: Rav came to Bavel and saw them reciting Hallel on Rosh Chodesh. He thought that he should stop them. Once he heard that they skip, he realized that it is a mere custom from their fathers.

(Beraisa): An individual should not start. If he started, he finishes.

Rishonim

Rif (Pesachim 26b): The custom is like Rav Yehudah.

R. Yonah (Berachos 7b DH v'Rabbeinu): R. Tam says that even on days that we do not finish Hallel, an individual must bless. He learns from the Isur to talk in the middle. The Gemara says that an individual may not begin. It does not discuss Hallel of Rosh Chodesh or similar matters. Rather, Hallel was enacted so that Tzaros not come on the Tzibur. When all Yisrael had an affliction, and Hash-m did a miracle and saved them, they would make a Yom Tov and say Hallel. They were obligated to bless, even though they did not finish. The primary enactment was only for a miracle for all of Yisrael - "v'Chi Savo'u Milchamah..." There was no enactment to say Hallel for a miracle for an individual. An individual should not begin for a miracle done for him not amidst Yisrael. If he began with a Berachah, he finishes and blesses. Even a Tzibur is called an individual, if all of Yisrael were not there. Bahag says so. 'Days when an individual finishes (or does not finish) Hallel' means that the Tzibur finishes or does not finish. An individual and Tzibur are the same. Just like a Tzibur blesses when we do not finish, also an individual. All this is difficult. 'Individual' does not refer to a Tzibur! An individual does not bless when we do not say full Hallel.

Rosh (ibid.): The Halachah follows Rabah. Bahag says that 'individual' is even a Tzibur, unless all of Yisrael are gathered, like when offering Pesach. Nevi'im enacted Hallel for when all of Yisrael are redeemed. When an individual says full Hallel, it is like Kri'as Shma for interrupting.

Rosh (Pesachim 10:32): We cannot derive from our Gemara that one must bless Birkas ha'Mazon over a cup. Perhaps the four cups were not enacted for an individual! Medrash Tehilim says that three must recite Hallel, so one can say Hodu to the others. If you would not say so, that one must pursue a Zimun, rather, that even an individual requires a cup for Birkas ha'Mazon, why do we discuss four cups? We always require one for Kidush and a second for Birkas ha'Mazon! Rather, normally we need not pursue a Zimun (and we may say Birkas ha'Mazon without a cup). The custom is not to pursue a Zimun. One can fulfill the Medrash with his wife and a child who reached Chinuch (training in Mitzvos). Even if only two cups are extra, we can say that it was enacted to drink four cups in all.

Poskim

Rema (OC 422:2): Some say that when an individual recites Hallel on Rosh Chodesh, he says to two to say with him the Roshei (beginnings of the) Roshei Perakim, and this is like b'Rabim. This is the custom for Hodu, but not for Ana.

Magen Avraham (7): Lechem Chamudos says that one says to two others only when we skip, due to Safek Berachos. The custom to do so even when we say full Hallel is mistaken. He did not see the Agur and Shibolei ha'Leket, who say that we say "Hodu" only to two (we ask them to praise Hash-m). On Seder night we strive to get three to say Hallel together, even without a Safek Berachah. The Mordechai says that it is a Mitzvah to answer the start of each Perek. Why is this unlike other Hodu's, e.g. in Hallel ha'Gadol (Tehilim 136)? Perhaps since here is like Shirah, others must answer, like we find regarding Shiras ha'Yam.

Shulchan Aruch (7): The Mitzvah is to say Hallel while standing.

Beis Yosef (DH Kasuv): Shibolei ha'Leket says that on Pesach night we say it sitting because we divide it (part before the meal, and part after), therefore we do not burden people to stand (twice). Also, he stands at the table holding his cup. Oh he will be distracted and the cup will fall. Maharam writes that on Pesach we recline like free men, therefore we do not exert people to stand.

Shulchan Aruch (479:1): It is a Mitzvah to pursue a Zimun (at the Seder).

Taz (3): The Tur brings from Medrash Tehilim that we require three, so one says "Hodu" to two. The Rosh says that this is not the custom. He tried to prove that one must pursue a Zimun to obligate a cup for Birkas ha'Mazon and there will be four in all, but rejected this, for perhaps the Mishnah teaches that in all there are four. The Tur teaches that one must pursue a Zimun. This is for Hodu. The Tur (182) rules that Birkas ha'Mazon requires a cup even for an individual! The third just answers to Hodu, and returns to his house. Also the Shulchan Aruch means like this. Even though the Rosh says that for three it suffices to say to his wife and a child who reached Chinuch, it is better with three men. If someone from another house came for a Zimun for Birkas ha'Mazon, and he eats the Afikoman here, after Birkas ha'Mazon he may stay here or return to his house. He should not do so unless there is a Zimun in his house without him.

Magen Avraham (2): The Tur says that regarding Hallel, it suffices for his wife and children to answer. This is like Tosfos (Sukah 38a DH Mi), who say that this Hallel is obligatory for women.

Rema: The greatest person present says Hodu and Ana, and the others answer. The Gadol can give permission to someone else to lead. They can join for Hallel, even if they did not eat together.

Magen Avraham (2): The Maharal says that a child may not say Hodu (for the others), but he may say Ana, for the others answer Ana (they need not be Yotzei through him). Mahariyo says that the Gemara curses one who makes a child or woman the Shali'ach for this, i.e. for Roshei Perakim. If there are only two, each says Hodu. The Agur says that the Mitzvah is with three men. Surely, l'Chatchilah one should do the ideal Mitzvah, but we are not so particular, since women join for this.

Mishnah Berurah (9): Chak Yakov says that since the custom is to sing Hodu, a woman should not say it. It is better to bless in one's house than to seek a Zimun for Birkas ha'Mazon in another house. One can join a Zimun for Hallel without eating there. If one wants to have a Zimun for Birkas ha'Mazon, he can invite guests to his Seder!

Mishnah Berurah (10): It is the ideal Mitzvah that one says Hodu and Ana and the others answer. It is not Me'akev.

Mishnah Berurah (12): For Ana one can let a child lead, so he will stay awake or for Chinuch, but not for Hodu.