The Levush (OC 685:1) writes that Adar Rishon follows the same zodiac sign of Shevat, and thus has no extra good mazel . The Kedushas Levi (Ki Sisa), however, writes that the twelve months correspond to the twelve Shevatim, with Adar corresponding to Yosef. Thus, in a leap year, the two Adar’s correspond to his sons, Ephraim and Menashe who are both under the same zodiac sign and subject to extra mazel (see Nitei Gavriel, Purim Katan p9). Thus, one should increase his level of simcha during Adar Rishon, too.
While many rishonim (Ran; Rosh, Nedarim 63a) write that when people talk about Adar during a leap year they are generally referring to Adar Rishon, the Mishna Berura (427:3) writes that for the sake of clarity the Chazan should ideally announce Adar Rishon during Mevorchim Hachodesh.
The Rema (OC 55:10) paskens that a boy born in Adar in a non-leap year must wait until Adar Sheni to be considered Bar Mitzva as only then is he thirteen halachic years old.
When one’s relative passes away during one of the Adar’s in a leap year, the yahrzeit is observed during that same month in a leap year; either Adar Rishon or Adar Sheni. When one’s relative passes away during Adar in a non-leap year, however, there is a machlokes as to which month the yahrzeit should be observed in. The Shulchan Aruch (OC 568:7) writes that it should be observed in Adar Sheni as that is the real Adar (Ridvaz 1:150). The Rema (YD 402:12) disagrees, writing that it should be observed in Adar Rishon, following the principle of ein ma’avirin al hamitzvos, we observe Mitzvos at the first opportunity, though concedes (OC 568:7) that as there is a machlokes, it is ideal to observe both (See Mishna Berura 568:42).