Note: This Q&A is adapted from an essay of R' Chananya Berman of Queens [NY] Kollel Boker
Q: I know that this year there are two months of Adar. Which is the real one?
A: The simple answer is, the second one: This is when we celebrate Purim. For this reason, if one was born in a leap year, his or her bar or bat mitzvah would take place in the Adar in which he or she was born. Of course if there is only one Adar in his or her bar/bat mitzvah year, the event would be celebrated in that month. (If you think about it, the possibility therefore exists for a boy born in Adar II to reach the age of Mitzvah obligation 13 years later before a boy born a few weeks earlier.) It is interesting that Rabbi Moshe Shternbuch (London-born, now an eminent senior rabbi in Jerusalem) writes that if one is born in Adar of a non-leap year he should observe his bar mitzvah as a stringency in Adar I and then again in Adar II. (He does not address the question of an additional party…)
Yahrtzeit: If the deceased passed away in Adar of a leap year the yahrtzeit would be observed in that month in future years. However if s/he died in Adar in a non-leap year, the Rema maintains that the yahrtzeit should be observed in the first Adar, out of respect. The Shulkhan Arukh upholds the standard preference for Adar II. The Rema also adduces an opinion that one should observe both days as a yahrtzeit. Indeed, there are many who do so, as a gesture of filial respect.
“Increasing joy” and Purim Katan: There is a discussion among the authorities whether there is a precept of increased simcha in Adar I, or is it only Adar II (in which Purim is celebrated). Rav Yaakov Emden holds that there is no specific idea of simcha, the Hatam Sofer in a responsum agreed with this idea. However, in an unrelated teshuva he signs off by saying "The first of Adar I, the month that we increase simcha" (H.M. 20). The Mishna Berurah (686:8) is not explicit but has been understood to endorse increased joy. Hey, why not? The Rema mentions the custom of having a festive meal on Purim Katan as well.