Why do we add an extra month to the year, and why is it specifically Adar?
Based on the verse in Parshas Re'eh (16:1), we know that Pesach must be during the spring.
Here's the problem: We follow the lunar calendar, and each month has a 29.5-day cycle, equaling 354 days a year, while a solar cycle has 365 days per year. So, each lunar year is 11 days earlier than the solar year.
To ensure that Pesach falls out during the right season (based on the solar cycle), we add another month seven times every 19 years. But why do we add the extra month to Adar? Why don't we, for example, have a double Shevat?
Rashi in tractate Rosh Hashanah (7a) explains, based on the verse in Re'eh (16:1), that the added month must be the closest to the spring.
Tosafos in tractate Sanhedrin (12a) tells us that since we have verses that refer to Adar as the "twelfth month," we can't insert any month before it; otherwise, during leap years, it would be the thirteenth month.
The Avudraham (Purim prayers) quotes the Mechilta (Bo:2) that just like we add a 30th day to some months- at the end of the month, we add a month to some years at the end of the year.