The obligation to recite Hallel on Rosh Chodesh:
On Rosh Chodesh one recites the abridged Hallel [i.e. half Hallel]. This applies for both a congregation and a person who is praying in private. The paragraphs of Lo Lanu and Ahavti are skipped.
Obligation or custom: The recital of Hallel on Rosh Chodesh is not required from the letter of the law. Nevertheless, it is the custom to recite it. Today, there is no man amongst Israel who does not read Hallel on Rosh Chodesh, Heaven Forefend.
Why does one recite Hallel on Rosh Chodesh? This custom is derived from the last Psalm in Tehillim 150 which mentions the word Hallelukah 12 times in correspondence to the 12 months. The last verse of Kol Haneshamah is repeated in correspondence to the 13th month of a leap year.
Why is there an abridged Hallel on Rosh Chodesh? Being that the recital of Hallel on Rosh Chodesh is only a custom, we therefore skip parts of it in order to emphasize its lack of obligation. Alternatively, the reason is because Rosh Chodesh is a day of atonement for children who would die of the Askara disease. The sacrifice of Rosh Chodesh would atone for them. Now, since this is a day of atonement and forgiveness just like Rosh Hashanah and Yom Kippur, therefore we do not recite the complete Hallel, just like Hallel is not recited on Rosh Hashanah.
Are women obligated to recite Hallel?
What is one to do if he accidently began to say the paragraph of Lo Lanu or Ahavti?
He is to conclude the verse in which he is holding and then continue with the next paragraph.
What is one to do if he already read the paragraph of Lo Lanu or Ahavti?
Seemingly, if he remembers in middle of Hallel, then in such a case he is to skip another paragraph, or verse, in order to emphasize that this Hallel is only due to custom. If, however, he already completed the Hallel, then there is nothing to do.
Sparks of Chassidus
Why we say Hallel on Rosh Chodesh:
The saying of Hallel is not just mere lip service but actually affects the world’s reception of Divine light. The prayer of Hallel is meant to draw down levels of above nature into the natural order of the world. Now, the weekday is meant to follow the natural order of the world and therefore is not fit for the recital of Hallel which represents miracle and above nature. Therefore, not only do we not recite Hallel during the week, but it is even forbidden according to Halacha, as stated above. On Shabbos, Hallel is likewise not recited being that Shabbos follows the order of above nature, and hence has no need for the above nature effects of Hallel. Rosh Chodesh however is a day in which work is optional, and thus, while not following the natural order of the world which requires one to work, is also not on the level of Shabbos. It is on this day that we say Hallel to draw down the light of above nature into nature.
 Michaber 422:2; Tur 422; Rambam Chanukah 3:8; Taanis 28b; See Admur 488:3; 490:7
 Michaber ibid
Opinion of Rambam: The Rambam [Chanukah 3:7] rules that the abridged Hallel was only established to be recited by a congregation, while an individual is not to recite it. However, if he began to recite it then he is to complete it.
 Siddur Admur; Tur 422; Rambam Chanukah 3:8; Elya Raba 422:12; Binyamin Zev 361; See Hashlama Taanis 28b; Hamichtam Sukkah 44b; Koveitz Moriyah 34:4 Elul 5775 in great length; See Biur Halacha 422:6 “Lemafreia” based on Rambam Chanukah 3:8 that in truth one can skip which area he desires, however the above areas of skipping is the accepted custom.
The reason: As for the reason that specifically these paragraphs are skipped, see Binyamin Zev ibid; Pirushei Siuddur Hatefila Larokeiach p. 627; Koveitz Moriyah ibid
Other customs: For other customs of where to skip, see Rambam ibid, Maharitz Geios, Siddur Rasag
 See Erechin 10b; Levush 422:2 that only by a Chag is there an obligation
 Admur 490:7; See Taanis 28b
 Piskeiy Dinim 2:6
 Shivlei Haleket in name of Geonim; Beis Yosef; Kaf Hachaim 422:22
 See Admur 488:3; 490:7; Rif Shabbos 11a; Abudarham
 Levush 422:3; Kaf Hachaim 422:23
 Admur 106:2; 479:7; M”A 422:5; Mishnah Sukkah 38a; Tosafus Brachos 20b; Kaf Hachaim 422:28; See Admur in Kuntres Acharon 296:3
The reason: As it is a Rabbinical time dependent Mitzvah. [Admur ibid; M”A ibid]
 Kesher Gudal 23:7; Beis Oved 16
 Rav A”C Naah in Yagdil Torah 4:1676
 The reason: As one may not say half of a verse. [ibid]
 Avnei Yashfeh O.C. 83:1; See Biur Halacha 422:6 “Lemafreia” based on Rambam Chanukah 3:8 that in truth one can skip which area he desires, however the above areas of skipping is the accepted custom. Accordingly, if one did not skip according to the accepted custom, he should skip another verse or paragraph.
 See Rambam ibid that the entire reason for saying half Hallel is to show that it is a mere custom, and hence Bedieved if the whole Hallel was said, it should not invalidate the Hallel.
 Sichas Chayeh Sara Mevarchim Hachodesh Kisleiv 1951