The Hebrew Months
The Jewish calendar is primarily lunar, with each month beginning on the new moon, 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.
​
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 II is added in the 3rd, 6th, 8th, 11th, 14th, 17th and 19th years of the cycle. The new year that began Monday, September 25, 1995 (Jewish calendar year 5756) was the 18th year of the cycle. Jewish year 5758 (beginning October 2, 1997) will be the first year of the next cycle.
​
In addition, Yom Kippur should not fall adjacent to a Sabbath, because this would cause difficulties in coordinating the fast with the Sabbath, and Hoshanah Rabba should not fall on Saturday because it would interfere with the holiday's observances. A day is added to the month of Heshvan or subtracted from the month of Kislev of the previous year to prevent these things from happening.
​
Adapted from http://www.jewishvirtuallibrary.org/the-jewish-calendar
