Halakhah הֲלָכָה

The month of אֲדָר שֵׁנִי (Adar Sheni, second Adar) begins this week!

Why do we have a leap month?
There are twelve months in the Jewish calendar, but they are based on the moon, so they are a little shorter than the months of the non-Jewish, solar calendar, which is based on the sun. In a normal year, the Jewish calendar is eleven days shorter than the solar calendar. But because seasons are caused by the sun (not the moon), this means that Jewish holidays would end up being eleven days earlier every single year, and, eventually, we would have Pesah in the winter and Hanukkah in the summer!
To prevent this from happening, the Jewish calendar includes a leap year by adding a whole month in seven out of every nineteen years. In those years, there are two months of Adar: the first one is called אֲדָר רִאשׁוֹן (Adar Rishon, first Adar), and the second one is called אֲדָר שֵׁנִי (Adar Sheni, second Adar).
Which month is Purim?
There is a מַחְלוֹקֶת (mahloket, disagreement) about this in the Talmud (Megillah 6b). According to one opinion, we should celebrate Purim during Adar Rishon since we should not pass up the opportunity to do a mitzvah, that is, by waiting until Adar Sheni. However, the halakhah is to celebrate Purim in Adar Sheni, to keep Purim and Pesah close together because both of them celebrate God saving the Jewish people. The 14th day of Adar Rishon is known as פּוּרִים קָטָן (Purim Katan, little Purim), and some people have a little celebration on that day, such as enjoying a more festive meal than normal (Rema 697:1).
When is my birthday?
If you were born during Adar in a normal year, then it is celebrated during Adar Sheni in a leap year. This means that your bat/bar mitzvah would be in Adar Sheni (Rema on Shulhan Arukh Orah Hayyim 55:10).
Two months of joy!
The Mishnah says that the only difference between the two months of Adar is reading Megillat Esther and giving gifts for the poor (these are the mitzvot of Purim, which is in Adar Sheni). In all other ways, the two months are the same. The Talmud (Ta’anit 29a) teaches that מִשֶּׁנִּכְנַס אֲדָר מַרְבִּין בְּשִׂמְחָה (when Adar comes, we increase our joy). Based on this mishnah, the entire two months of Adar have this extra joy!