Why is Rosh Chodesh sometimes one day and sometimes two?

The Jewish calendar is based on the lunar cycle. Since a lunar month is approximately 29 days and twelve hours, we alternate months — one month is twenty-nine days and the next month is thirty. When the Sanhedrin (Rabbinical Supreme Court) was convened, the months were determined by witnesses who testified that they saw the crescent new-moon. The Sanhedrin would assemble on the thirtieth of each month, for perhaps witnesses would come and this day would be designated Rosh Chodesh ("Head of the Month") of the upcoming month (rendering the previous month a 29 day month).

Since the thirtieth day of the month was always potentially Rosh Chodesh, whenever a month has thirty days, the thirtieth day is observed as Rosh Chodesh together with the next day, the first of the following month.

However, if a month has only twenty-nine days, then the Rosh Chodesh of the following month will be only one day--the first of the month.

The following months always have two days of Rosh Chodesh (the first day of the month plus the last day of the previous month): Cheshvan, Adar (and Adar II), Iyar, Tammuz, and Elul.

The following months always have one day of Rosh Chodesh: Tishrei,1 Shevat, Nisan, Sivan, and Av.

The months of Kislev and Tevet fluctuate; some years they both have one day of Rosh Chodesh, some years both have two days, and some years Kislev has one day and Tevet has two days Rosh Chodesh.