Rosh Chodesh, the beginning of the Jewish (lunar) month, is a minor holiday of one or two days, depending on whether the previous month was 29 or 30 days.  When the preceding month is 30 days long, the 30th day becomes the first day of Rosh Chodesh and the second day of Rosh Chodesh is the first day of the succeeding month.
Rosh Chodesh used to be officially declared in Jerusalem each month by the Sanhedrin, based on evidence from at least two witnesses who had seen the new moon in the western sky.  For the past 1700 years or so, the new moon (and the dates for the entire Jewish calendar) have been determined by a formula prescribed by Hillel HaNasi (then head of the Jewish community in Eretz Yisrael).