The very first commandment that was given to the entire Jewish nation was the commandment to celebrate the appearance of the new moon. “This month shall be unto you the beginning of months; it shall be the first month of the year to you” (Exodus 12:2). That first month was the month of Nisan, the month of Passover.
The commandment of celebrating the appearance of the new moon was the first step in the Divine plan to teach the Jewish people to be free and to be holy. By instructing the Children of Israel to mark the start of each month, to take charge of their calendars, God gave them a sense of controlling time--something that they had lacked as slaves.
Originally, Rosh Chodesh (the "head" of the month) was determined by the testimony before the Sanhedrin of witnesses that had seen the new moon. Once the testimony was accepted, hilltop fires were lit to announce the new month to neighboring communities, who spread the word in the same manner. In the year 358 C.E., however, a set lunar calendar, integrated with the solar calendar, was introduced and accepted in lieu of witnesses.
Rosh Chodesh is celebrated with the addition of Hallel (Psalms of praise), ya’aleh v’ya’vo and musaf to the prayers, as well as a general acknowledgment of it being a festive day (nicer clothes, nicer food, etc). In the times of the Temple, the day began with the sound of the shofar, and special sacrifices were offered in the Temple.
Rosh Chodesh is either one day or two, depending on the month. Rosh Chodesh Elul is celebrated for two days, the first of which is today.