The Megillat Esther is generally read in synagogues during the evening of Purim and the next morning. Where do we first come across this custom in our sacred literature? The gemara in Meseches Megillah 4a states (translation from Sefaria.org The William Davidson digital edition of the Koren Noe Talmud Bavli):

While the gemara is interpreting the word v’lishnota as “repeat”(compare is the word “mishnah” which also means “to repeat”), the root lamed-shin-nun-nun alludes to “talking”. This implies that not only is it a mitzvah to hear the megillah being read, the discussion implies that the megillah must be read aloud. We see this often with Torah study in which the matters being discussed (even alone) are read aloud.

With regard to the the reading of the megillah during a year in which a second Adar (Adar II) is added, the custom today (since Adar is the twelfth month in the Jewish calendar) is to celebrate Purim during Adar II. The Mishnah in Megillah 6b and gemara in Megillah 6b (translation from Sefaria.org The William Davidson digital edition of the Koren Noe Talmud Bavli) also discuss some common customs:

We see here, a discussion regarding the practice of Shalach Manot (Gifts to the poor). The month of Adar is also a time of increasing joy as we celebrate the miracle of Purim and prepare for Pesach and our liberation from bondage.

Chag Sameach Purim to all!