Rosh Chodesh is the monthly celebration of the New Moon, according to the Jewish calendar. The Jewish calendar follows lunar months, each with 29 or 30 days, although the year is solar. Some scholars believe that lunar months derive from ancient nomadic calendars and solar years are the invention of agricultural societies; the Jewish calendar combines the two. Many Jewish festivals are tied to the lunar cycle; for example Sukkot and Passover begin on the full moon, in the middle of the month. Since 12 lunar months do not add up to one complete solar year, additional "leap months" are intercalated into the calendar in seven years out of a 19-year cycle.
Long ago, the appearance of the new moon each month was attested by witnesses. Once their testimony was deemed credible, fires were set on the hilltops to announce the new month to neighboring communities who, in turn, passed the message along. This system proved both dangerous and cumbersome, and once Jews lived outside Eretz Yisrael, it was wholely inadequate.
The present Jewish calendar was introduced in the time of Hillel II (358/9 CE), at which time astronomical calculations replaced the practice of calling witnesses before the Sanhedrin. Since that time, it has been possible to calculate the Jewish calendar well into the future on the basis of scientific calculations.
Sometimes Rosh Chodesh is one day, but sometimes it is celebrated for two days. Months are based on the lunar cycle, of course. The amount of time required for the moon to make one complete revolution around the earth is determined by the conjunction of the earth, moon, and sun (i.e. they lie along a line). When this happens, it is the new moon. A complete revolution takes 29 days, 12 hours, 44 minutes, 3-1/2 seconds. Days in our calendar must begin at sundown, regardless of when the conjunction actually takes place. Hence it is necessary to either add or subtract a half a day from each calendar month. For this reason, Hebrew months alternate between 29 and 30 days in length, with the actually moment of the new moon falling in between. The 30-day month is called "malei" (full) and the 29-day month is called "chaser" (defective). In any given leap year, Nisan, Sivan, Av, Tishrei, Shevat and Adar I are malei; Iyar, Tammuz, Elul, Tevet, Adar, and Adar II are chaser. Cheshvan and Kislev are sometimes malei and sometimes chaser. When a month is 30 days in length, the following month Rosh Chodesh is celebrated for two days because the 30th day of the month past is counted as Rosh Chodesh and the first day of the subsequent month as the second day of Rosh Chodesh. In this particular year, for example, last October 20-21 was Rosh Chodesh. Oct. 20 was 30 Tishrei and Oct. 21 was 1 Cheshvan. It happens again this month: November 19-20 will be celebrated as Rosh Chodesh. Nov. 19 is 30 Cheshvan and Nov. 20 is 1 Kislev. Nisan, Sivan, Av, and Tishrei always begin with one day of Rosh Chodesh; Iyar, Tammuz, Elul, Cheshvan, Adar I, and Adar II always begin with two days Rosh Chodesh. Kislev and Tevet vary between one and two days Rosh Chodesh.
Rosh Chodesh has long been recognized as a women's holiday. In the Talmud [tractate Megillah 22b], we read that women are exempt from work on Rosh Chodesh. Rashi, on commenting on this passage, delineates the activities from which they may refrain: spinning, weaving, and sewing, because these are the skills which women so enthusiastically contributed to the building of the Mishkan (Tabernacle). Why do women merit a special holiday once a month? In midrash Pirke DeRabbi Eliezer, chapter 45, we are told that in the incident of the Golden Calf, the women refused to relinquish their earrings to the men who were building the calf. As a reward, God gave them an extra holy day each month, free from work. It is customary to wear new clothing on Rosh Chodesh, in celebration of the day's special character.
Rosh Chodesh is announced on the Shabbat prior with a special benediction recited during the Torah Service. Rosh Chodesh, itself, is celebrated with a partial Hallel, musaf (in remembrance of the extra sacrifice brought on Rosh Chodesh), and Ya'aleh V'yavo is added to the Amidah and Birkhat HaMazon. In addition, the Haftarah for Rosh Chodesh falling on Shabbat is from Isaiah 66, a passage which employs fertility imagery to describe God and Zion as life-bearers, providing nurturance to the people Israel; the passage further prophesies a special pilgrimage to Jerusalem on Rosh Chodesh in the future.
Sources: Judaism 101