Q: This year is a Jewish leap year, there are two months of Adar. Can you explain why we have Jewish leap years and when in Jewish History it was instituted?
A: Unlike Christianity, which uses a solar calendar and Islam, which uses a lunar calendar, Judaism uses a luni-solar calendar, which has features of both! This means that we celebrate the beginning of each lunar month and festivals fall according the lunar cycle. However, the Torah also requires that Pesach falls in the spring and Succos in the autumn. This means that a mechanism is required to ensure that the lunar and solar calendars remain roughly in harmony with each other (a 12-month lunar year is approximately 11 days shorter than a solar year). This process is called intercalation - the addition of a whole lunar month to the calendar from time to time.
In ancient times, the rabbis would work out on a yearly basis how close Pesach would fall to the vernal equinox and if necessary, add a month in the spring before Pesach. These days, we have a fixed calendar of 19-year cycles, of which seven include an extra, thirteenth, month.
In a regular year, the last lunar month in the cycle is called Adar; in a leap year, there are two Adars, known as Adar I and Adar II. Technically, Adar I is the added month, so Purim falls in Adar II.