Gemara

(R. Yochanan): 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 - Question: We should say full Hallel on Rosh Chodesh, for it is called Mo'ed!

Answer: One may do Melachah on Rosh Chodesh;

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

Ta'anis 28b - Question: Why doesn't the Mishnah say that there is no Ma'amad on Rosh Chodesh Nisan, because there is Hallel?

Answer (Rava): This teaches that Hallel of Rosh Chodesh is not mid'Oraisa;

It is not among the days that R. Yochanan listed.

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.

Berachos 14a (Beraisa): In between Perakim of Shma, one may greet his Rebbi, and we need not say that he responds;

In the middle, he greets only one whom he fears.

(Rabah): On days when an individual says the full Hallel, he may interrupt between Perakim, but not in the middle of a Perek;

On days when an individual says the abridged Hallel, he may interrupt even in the middle of a Perek.

Question: Rav bar Sheva visited Ravina on a day when an individual says partial Hallel, and Ravina did not interrupt!

Answer: Ravina did not consider Rav bar Sheva so important that he must greet him.

Shabbos 118b: One who says Hallel every day blasphemes.

Sukah 39a (Mishnah): Where the custom is to bless on Hallel, we bless.

(Abaye): This refers to afterwards, but it is a Mitzvah to bless beforehand.

(Rav Yehudah): We bless on all Mitzvos beforehand.

Rishonim

Rif (Shabbos 11b): Hallel of Rosh Chodesh is not mid'Oraisa. It is a mere custom. Therefore, we do not finish it. Rav saw people in Bavel skip in Hallel. He realized that it is a mere custom. A Beraisa teaches that an individual should not start. If he started, he finishes. Therefore, if an individual wants to recite Hallel on Rosh Chodesh, he recites without a Berachah and skips.

Ran (DH Tana): It depends on the custom, i.e. for a Tzibur. A Beraisa teaches that an individual, even in Bavel, may not (begin). If he started, he finishes. A Tzibur starts even l'Chatchilah. Others say that even for Bnei Eretz Yisrael, since they do not have the custom (to say it), an individual or Tzibur does not begin. However, if he started, he finishes. The Rif connotes that it discusses Bnei Bavel. All our customs follow Bnei Bavel.

R. Yonah (Berachos 7b DH v'Nir'eh): It seems that even on days that we do not finish Hallel, an individual blesses on it. One may interrupt in the middle of a Perek only for one whom one fears or must honor, but not for other talk. If one does not bless, since it is mere recital, one should be allowed to interrupt for anything! Rav did not realize that it is only a custom until they skipped, but not from the text of the Berachah "to recite Hallel", and not 'to finish Hallel." This is because they always used to bless "to recite." Later, people distinguished (which Berachah to say).

Rosh (Berachos 2:5): They started to bless 'to finish Hallel' and 'to recite Hallel' to hint to the Tzibur when we finish and when we skip.

R. Yonah (ibid.): However, the Rif explicitly says that an individual does not bless. He explains the Beraisa to teach that an individual does not begin with a Berachah, but if he began with a Berachah, he finishes with a Berachah, i.e. Baruch Atah Hash-m Melech Mehulal ba'Tishbachos. If it is without a Berachah, why may an individual not begin? Is it forbidden to say verses?! Here, the Gemara forbids interrupting for mere talk, but not because he blessed. Rather, even though he did not bless, once one begins the praise of Hash-m, he should not interrupt for mere talk.

Rambam (Hilchos Chanukah 3:7): On Rosh Chodesh, reciting Hallel is a custom. It is not a Mitzvah. The custom is in a Tzibur, therefore we skip. We do not bless, for we do not bless on customs. An individual does not recite at all. If he began, he completes it, and skips like the Tzibur does. On Pesach (after the first Yom Tov) we recite with skipping, like on Rosh Chodesh.

Magid Mishneh: Our custom is that a Tzibur blesses on it, and an individual says it without a Berachah.

Rebuttal (Ra'avad): This is wrong, like I explained (Hilchos Berachos 11:16, that we bless on important customs.). If an individual began, he recites as he desires (with or without skipping), and blesses at the end like the Tzibur.

R. Yonah (ibid.): The Rambam says that even a Tzibur does not bless on Rosh Chodesh, for it is only a custom. We do not bless on customs, like it says in Sukah (44b). The one who says that we do not bless on banging Aravos holds that it is only a custom. The Gemara explicitly says that a Tzibur blesses! An individual may not begin, i.e. with a Berachah. This implies that a Tzibur must begin with a Berachah! The Rif says that an individual recites without a Berachah, which connotes that a Tzibur must bless. However, this is not difficult for R. Natrunai Ga'on and Rav Hai Gaon. They hold that an individual does not begin, i.e. he need not begin at all, even without a Berachah. If he erred and began with a Berachah, he finishes until Yehalelucha, without a closing Berachah. However, Yom Tov Sheni is only a custom, since we know when the months are fixed. Why do we bless on Hallel of Yom Tov Sheni? The Rambam's proof from Aravah is weak. Not all customs are the same. We merely bang the Aravah, so we do not bless on it. Hallel has praise of Hash-m, so we bless on it.

Rosh (ibid.): The Halachah follows Rabah. When an individual says full Hallel, it is like Kri'as Shma for interrupting. When an individual does not full Hallel, one greets for honor, and answers anyone. If not, one does not interrupt. R. Tam derives that one blesses, for if not, this is like reciting Tehilim, and 'interruption' does not apply. This is unlike people say about Rashi, that he did not bless on Rosh Chodesh, for it is only a custom. Yom Tov Sheni is only a custom, yet we bless on Shofar and Matzah on Yom Tov Sheni. The Yerushalmi says that if the Halachah wavers, see what the Tzibur does. The Tzibur blesses. Bahag rejected the Rif's proof; 'individual' is even a Tzibur.

Tosfos (10b DH Yud): One should follow R. Tam. Even an individual blesses also beforehand. Even though he is not obligated to say Hallel, once he decides to do so, it is not a Berachah Levatalah, just like women bless on the Lulav even though they are exempt.

Poskim

Shulchan Aruch (OC 422:2): On Rosh Chodesh we say partial Hallel, whether alone or in a Tzibur.

Beis Yosef (DH u'Mah she'Chasav v'Korin): Shibolei ha'Leket (172) brings from the Ge'onim a hint from Tehilim 150 that we say Hallel on Rosh Chodesh. It says Hallelu 12 times, corresponding to 12 months. Therefore, we repeat (the last verse) "Kol ha'Neshamah" for a leap year, which has 13 months, and for a (Rosh) Chodesh that has two days.

Kaf ha'Chayim (22): Even though one of the 12 is Rosh Hashanah, it is also Rosh Chodesh, and we would say Hallel if it were not the day of judgment.

Kaf ha'Chayim (23): The Levush says that we skip in Rosh Chodesh because it is a day of atonement, so it is like Rosh Hashanah and Yom Kipur, so we do not sing full Shirah.

Gra (DH Bein): Even though the Gemara said that an individual does not finish Hallel, the same applies to a Tzibur. Rav saw that they skipped. Even though we learned that an individual does not start, the custom is that they do. Even the Tzibur is not obligated; it is merely a custom (Tosfos Berachos 14a DH Yamim).

Magen Avraham (5): R. Yerucham says that if one was already Yotzei, he may not recite again for the sake of another, i.e. with a Berachah. Perhaps he forbids even without a Berachah, for one who says Hallel every day blasphemes. Therefore, one may recite it like Shirah only when Chachamim enacted.

Kaf ha'Chayim (34): There is a reason to say Hallel on Rosh Chodesh based on Kabalah.

Shulchan Aruch (ibid.): An individual does not bless on it. Some say that even a Tzibur does not bless on it before or after. This is the Rambam's opinion, and this is the custom in all Eretz Yisrael and surrounding lands.

Gra (DH veha'Yachid): Since an individual has no obligation to begin, he does not bless.

Rema: Some say that also an individual blesses on it. This is the custom in these lands. In any case one should be careful to recite in the Tzibur, in order to bless on it with the Tzibur.

Beis Yosef (DH Kasav): Kol Bo says in the name of R. Peretz that it is good to say it with the Tzibur, even before Tefilah. The Roke'ach saw his Rebbi recite it without the Tzibur, sitting, during Kri'as ha'Torah.

Mishnah Berurah (16): To be concerned for this opinion, if one was late to the Beis ha'Keneses, he should say Hallel with the Tzibur, and pray afterwards. This applies when we skip.

Rema: Some say that when an individual recites it, he says to two to say with him the Roshei (beginnings of the) 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.

Gra (DH v'Yesh): Three is called Rabim, like it says in Gitin 46a.

Gra (DH v'Nahagu): Hodu itself is called 'Roshei Perakim.' We learn from Sukah 38b (it says 'he says Hodu la'Shem, and they answer Hodu la'Shem. From here we learn that it is a Mitzvah to answer the Roshei Perakim.

Mishnah Berurah (17): They answer Hodu and Ana.

Mishnah Berurah (18): If one does not find another two, he need not pursue them.