The Talmud (Erechin 10B) explains that while Rosh Chodesh is called a moed still Hallel is not recited on Rosh Chodesh since there is no prohibition of work on Rosh Chodesh. However, the custom has developed to say Hallel on Rosh Chodesh, though some verses are skipped to indicate that it is just a custom. This custom existed at least from the time of the Talmud. The Talmud (Ta'anit 28b) records that Rav was visiting Bavel and he saw the people reciting Hallel on Rosh Chodesh. Rav wanted to stop the congregation from reciting Hallel, but when he saw that they skipped verses (the so called "half" Hallel), he said it was ok because he realized that they were only reciting Hallel as a custom.
Why did the custom develop to recite the "half" Hallel on Rosh Chodesh? Many reasons have been suggested. Rav Zevin (1956, p.146) quotes the Emek Halacha that the custom developed since there were some customs to limit working on Rosh Chodesh. Arukh Hashulchan (422:6) suggests that the reason is that the re-appearance of the moon signals the Jewish people's survival. Rav Schachter (1994, p. 194) quotes that R. Moshe Soloveitchik suggested that in the Bet ha-Mikdash, they used to say a full Hallel on Rosh Chodesh, and hence afterwards there developed a custom to say a partial Hallel. The simplest explanation is mentioned by Shinan (1999, p. 198) that Rosh Chodesh was/is considered a festive day, and the people wanted to mark the day by reciting Hallel.
The Talmud (Ta'anit 28b) did not record which verses were skipped, and different customs have developed as to which verses to skip. Today, I believe the universal custom is to skip the first ten verses in Tehillim chapters 115 and 116. The Rambam (Laws of Hanukkah 3:8) writes that his custom was also to skip chapter 117 (two verses) and the first four verses of chapter 118. The Maggid Mishnah (Spain, 14th century) in his comments on the Rambam, writes that the Rambam's custom was also the custom in Spain in his days.
Why were the first ten verses in Tehillim chapters 115 and 116 chosen to be skipped? This is not clear, especially since in one case the verses we skip 115:8-10 are connected to 115:11, which we recite. Monk (1978, p. 87) quotes from the Eliyahu Rabbah that these verses are similar to other verses from the same chapter that we do recite. Monk also quotes R. S. R. Hirsch that the verses skipped are cries from troubles. A variation on this is that maybe the skipped verses are considered "sadder." For example, 115:2 records that that the nations of the world will ask where is our G-d? Also, 116:3 records that the person is in the throes of death.
There are at least five opinions as to whether Hallel should be recited with a blessing on Rosh Chodesh. One, (Rambam, Laws of Hanukkah 3:7), is that the congregation recites Hallel without reciting any blessings, and the individual who prays alone does not recite Hallel at all.
Why did the custom develop to recite the "half" Hallel on Rosh Chodesh? Many reasons have been suggested. Rav Zevin (1956, p.146) quotes the Emek Halacha that the custom developed since there were some customs to limit working on Rosh Chodesh. Arukh Hashulchan (422:6) suggests that the reason is that the re-appearance of the moon signals the Jewish people's survival. Rav Schachter (1994, p. 194) quotes that R. Moshe Soloveitchik suggested that in the Bet ha-Mikdash, they used to say a full Hallel on Rosh Chodesh, and hence afterwards there developed a custom to say a partial Hallel. The simplest explanation is mentioned by Shinan (1999, p. 198) that Rosh Chodesh was/is considered a festive day, and the people wanted to mark the day by reciting Hallel.
The Talmud (Ta'anit 28b) did not record which verses were skipped, and different customs have developed as to which verses to skip. Today, I believe the universal custom is to skip the first ten verses in Tehillim chapters 115 and 116. The Rambam (Laws of Hanukkah 3:8) writes that his custom was also to skip chapter 117 (two verses) and the first four verses of chapter 118. The Maggid Mishnah (Spain, 14th century) in his comments on the Rambam, writes that the Rambam's custom was also the custom in Spain in his days.
Why were the first ten verses in Tehillim chapters 115 and 116 chosen to be skipped? This is not clear, especially since in one case the verses we skip 115:8-10 are connected to 115:11, which we recite. Monk (1978, p. 87) quotes from the Eliyahu Rabbah that these verses are similar to other verses from the same chapter that we do recite. Monk also quotes R. S. R. Hirsch that the verses skipped are cries from troubles. A variation on this is that maybe the skipped verses are considered "sadder." For example, 115:2 records that that the nations of the world will ask where is our G-d? Also, 116:3 records that the person is in the throes of death.
There are at least five opinions as to whether Hallel should be recited with a blessing on Rosh Chodesh. One, (Rambam, Laws of Hanukkah 3:7), is that the congregation recites Hallel without reciting any blessings, and the individual who prays alone does not recite Hallel at all.
Two (second opinion in the Tur, 422, Machzor Vitry quoted in Tosafot Berakhot 14a, Yamim? and probably the second opinion in Shulchan Arukh, 422:2) both the congregation and the individual recite Hallel without any blessings.
Three, (first opinion in Tur Orah Chayyim 422) the congregation recites Hallel with a blessing, but the individual does not recite Hallel at all.
Four, (Rif, Maggid Mishnah in his comments on the Rambam) the blessing is said when reciting Hallel with the congregation, but if praying alone, then a person recited Hallel without a blessing.
Five, (Rabbenu Tam, Tosafot Berakhot 14a, Yamim), both the individual and the congregation recite Hallel with a blessing.
There are two separate arguments here. The first argument is whether a person recites a blessing on a custom (opinions three, four and five) or no (opinions one and two). The second argument concerns the nature of Hallel. Opinions one, three and four view Hallel as a communal prayer, and hence the real or even only obligation is on the congregation and not the individual. However, opinions two and five, who are Ashkenazim, make no distinction between the congregation and the individual.
The Maggid Mishnah writes that the fourth approach was the custom in Spain his time (14th century). R. Yosef Caro (1488, Spain-1575, Israel) in his work the Bet Yosef (422) quotes the Maggid Mishnah's opinion, and my guess is that this was his practice at least until he came to the land of Israel (in 1536). In the Shulchan Arukh (Orah Chayyim 422:2) he adds that in Israel, the practice was to recite Hallel without a blessing on Rosh Chodesh when praying with a congregation, and it seems (the word sh-af) that the custom was also that an individual when praying alone would recite Hallel without a blessing. Apparently, when he wrote the Bet Yosef in Turkey, he did not know that the prevalent practice in Israel (Sefat?) was not to make the blessing by Hallel when praying with a congregation on Rosh Chodesh since he only mentioned this information about Israel in the Shulchan Arukh and not in the Bet Yosef. This has become the prevalent opinion amongst Sefardim in Israel, though I was told by two people that Sefardim from North Africa recite a blessing by Hallel on Rosh Chodesh.
A proof that the custom amongst Sefardim was to recite Hallel with a blessing when reciting Hallel together as a congregation on Rosh Chodesh is that the Shulchan Arukh 487:4 rules that one should recite Hallel with a blessing on the night of Pesach, and this is at best a custom, as it was not mentioned in the Talmud unlike Rosh Chodesh, see Bet Yosef 473, be-inyan.
A proof that the custom amongst Sefardim was to recite Hallel with a blessing when reciting Hallel together as a congregation on Rosh Chodesh is that the Shulchan Arukh 487:4 rules that one should recite Hallel with a blessing on the night of Pesach, and this is at best a custom, as it was not mentioned in the Talmud unlike Rosh Chodesh, see Bet Yosef 473, be-inyan.
Another example is by the lighting of the Hanukkah menorah in synagogue that the Shulchan Arukh (Orah Chayyim 671:7) records that one recites a blessing on this lighting even though it is a custom, apparently from the time of the Geonim. The Sharei Teshuvah (671:10) notes the incongruity between the Shulchan Arukh’s views on not reciting a blessing by Hallel on Rosh Chodesh and the reciting of blessings when lighting Hanukkah candles in the synagogue. Also see Mishnah Berurah, 671:44.
The Rama (422:2) in his comments on the Shulchan Arukh quotes opinion five to recite Hallel with a blessing whether one is reciting Hallel with the congregation or by himself. He writes that this was the custom in his time (16th century) amongst Ashkenazim, and this has remained the Ashkenazi practice even amongst Ashkenazim who pray according to nusach Sefard.
With regard to the blessing at the end of Hallel, the Mishnah (Sukkah 3:11, Talmud Sukkah 38a) seems to write that some people had the custom to recite the blessing on Hallel and some people did not have the custom. The Talmud (Sukkah 39a) quotes Abbaye that this comment in the Mishnah is only in reference to the blessing at the end of the Hallel, but before reciting the Hallel there is a commandment (by the full Hallel) to recite a blessing. The Rambam (Laws of Hanukkah 3:10) quotes this dictum, that the blessing afterwards is dependent on the custom of the place, but my understanding is that today whenever a blessing on Hallel is said beforehand, then a blessing is also recited after reciting Hallel.
A final question is whether the individual should recite the blessing after he hears the chazzan recite the blessing. The Bach (on the Tur, 422) writes that the individual should recite the blessing. However, R. Zevin (1956, p.146) quotes from the Ba'al ha-Tanya that the congregation should just say amen to the chazzan's blessing (both by the beginning and ending blessings) and not recite the blessings themselves. The Arukh Hashulchan (422:8) writes that this was also his practice since a person should not say extra blessings when there is an argument as to whether the blessings are necessary. Rav Schachter (1994, p.175) quotes Rav Soloveitchik that a person should recite the blessings before the chazzan recites the blessing, just as the Mishnah Berurah (619:3) suggests by the scheyanu on Yom Kippur. My impression is that most people follow the Bach and recite the first blessing of Hallel after the chazzan has already said the blessing.
Monk, Eliyahu, 1978, World of Prayers, Jerusalem: Mossad Harav Kook.
Schachter, Hershel, 1994, Nefesh Harav, New York: Flatbush Beth Hamedrosh
Shinan, Avigdor, 1999, Siddur Avi Chai, Jerusalem: Yediot Achronot and Sifri Hemed.
Zevin, Sholom Yosef (1888-1978), first published 1944, seventh edition 1956, Moadim be-hlakhah, Jerusalem.