“There is an opinion in the Talmud that Purim should be celebrated in Adar I based on the rule [that] ‘one does not ‘pass over’ or forego the performance of the mitzvahs’; rather, when presented with the opportunity to perform a mitzvah, one should do so immediately. Thus, it is reasoned that Purim should be celebrated at the first opportunity, i.e., Adar I.
Nevertheless, the halachah follows the opinion that Purim is celebrated in Adar II, for it is closest to the month of Nissan, and we want to ‘connect redemption to redemption,’ i.e., the redemption of Purim to the redemption of the Exodus, which is in the month of Nissan.
Although Purim is celebrated in Adar II, the days 14 and 15 Adar I are known as Purim Katan (‘Small Purim’) and Shushan Purim Katan (‘Small Shushan Purim‘) respectively. On these days, Tachanun (penitential prayers) are omitted, one doesn’t fast, and eulogies are generally not made.”