Rosh Chodesh which literally means "Head of the Month" is a monthly holiday that celebrates the arrival of the new moon, marking the start of a new month in the Jewish calendar.
The Jewish Calendar is a lunar calendar with twelve lunar months and 354 days per year. Each lunar month consists of 291/2 days. Since it is impossible to switch months in the middle of the day, certain Hebrew months have twenty-nine days whilst others have thirty days.
In order for the Jewish Festivals to be synchronized with the proper season, for example Passover (Pesach) which must fall in the spring (in Israel - the Northern Hemisphere), a leap year is added every few years. In a nineteen year cycle there will be seven leap years.
The Jewish leap year, however, is not like the leap year in the Gregorian calendar which is one day added on at the end of February every four years. A Jewish leap year adds on an entire month to close the gap between the 354 day lunar calendar and the 365 day solar calendar. Since the month of Adar (approximately February-March) is the last month of the Hebrew year, we add on an extra month at this time. We then have Adar I and Adar II which assures us that the holidays will be celebrated in the proper season.
Originally, the date of Rosh Chodesh was confirmed on the testimony of witnesses observing the new moon, a procedure known as Kiddush Hachodesh (sanctification of the month). After the Sanhedrin (an assembly of either 23 or 71 elders or rabbi's, appointed to sit as a tribunal in every city in ancient Israel) declared Rosh Chodesh for either a full 30 day month or a 29 day month, news of it would then be communicated throughout Israel and the diaspora.
This system was dependent on the functioning of the Sanhedrin to declare the month, and to communicate this month to far-flung Jewish communities. In the 4th century CE, this became impossible and instead a fixed calendar of 29 and 30 day months was instituted by Hillel.
At the end of a 29 day month, Rosh Chodesh is celebrated for one day, on the first day of the new month. At the end of a 30 day month, Rosh Chodesh is celebrated for two days - the 30th day of the previous month, and the 1st day of the new month.
With the exception of the month of Heshvan, the eighth month of the year, every single month of the Jewish Calendar has some sort of special day or special observance. The importance of Rosh Chodesh is therefore obvious.
The names of the months are not Hebrew but Babylonian. The Bible refers to the Jewish holidays only by the order of the months and not by the Babylonian names which were assigned long after the canonization of the Torah. For example, the Torah gives the date of Yom Kippur as the tenth day of the seventh month, whilst the date of Pesach is the fifteenth day of the first month.