(From the Jewish Virtual Library Website)
The Jewish calendar is based on three astronomical phenomena: the rotation of the Earth about its axis (a day); the revolution of the moon about the Earth (a month); and the revolution of the Earth about the sun (a year). These three phenomena are independent of each other, so there is no direct correlation between them. On average, the moon revolves around the Earth in about 29½ days. The Earth revolves around the sun in about 365¼ days, that is, about 12.4 lunar months.
The Jewish calendar, coordinates all three of these astronomical phenomena. Months are either 29 or 30 days, corresponding to the 29½-day lunar cycle. Years are either 12 or 13 months, corresponding to the 12.4 month solar cycle.
The lunar month on the Jewish calendar begins when the first sliver of moon becomes visible after the dark of the moon. In ancient times, the new months used to be determined by observation. When people observed the new moon, they would notify the Sanhedrin. When the Sanhedrin heard testimony from two independent, reliable eyewitnesses that the new moon occurred on a certain date, they would declare the Rosh Chodesh (first of the month) and send out messengers to tell people when the month began.
The problem with strictly lunar calendars is that there are approximately 12.4 lunar months in every solar year, so a 12-month lunar calendar loses about 11 days every year and a 13-month lunar gains about 19 days every year. The months on such a calendar "drift" relative to the solar year. On a 12 lunar month calendar, the month of Nissan, which is supposed to occur in the Spring, would occur 11 days earlier each year, eventually occurring in the Winter, the Fall, the Summer, and then the Spring again. To compensate for this drift, an extra month was occasionally added. The month of Nissan would occur 11 days earlier for two or three years, and then would jump forward 29 or 30 days, balancing out the drift. In ancient times, this month was also added by observation: the Sanhedrin observed the conditions of the weather, the crops and the livestock, and if these were not sufficiently advanced to be considered "spring," then the Sanhedrin inserted an additional month into the calendar to make sure that Pesach (Passover) would occur in the spring (it is, after all, referred to in the Torah as Chag he-Aviv, the Festival of Spring!).
A year with 13 months is referred to in Hebrew as Shanah Me'uberet (pronounced shah-NAH meh-oo-BEH-reht), literally: a pregnant year. In English, we commonly call it a leap year. The additional month is known as Adar I, Adar Rishon or Adar Alef. It is inserted before the regular month of Adar (known in such years as Adar II, Adar Sheini or Adar Beit). Note that Adar II is the "real" Adar, the one in which Purim is celebrated, the one in which yahreits for Adar are observed, the one in which a 13-year-old born in Adar becomes a Bar Mitzvah Adar I is the "extra" Adar.
In the fourth century, Hillel II established a fixed calendar based on mathematical and astronomical calculations. This calendar, still in use, standardized the length of months and the addition of months over the course of a 19 year cycle, so that the lunar calendar realigns with the solar years. Adar I is added in the 3rd, 6th, 8th, 11th, 14th, 17th and 19th years of the cycle. The current cycle began in Jewish year 5758 (the year that began October 2, 1997).
