Maseches Megillah, Chapter 1, Mishnah 4. This mishnah deals with a leap year, and with the question of in which of the two months of Adar the mitzvos of Purim apply.
The text of the mishnah:
"Karu es haMegillah baAdar haRishon v'nis'abrah hashanah - korin osah baAdar haSheini" - This refers to someone who read the Megillah on the fourteenth of Adar, assuming that this was an ordinary Adar, and afterward the year was declared a leap year, so that it turned out retroactively that this Adar was Adar Rishon. In such a case he reads the Megillah again in Adar Sheini, since Adar Sheini is the month in which Purim is celebrated.
"Ein bein Adar haRishon l'Adar haSheini ela:" - The only differences between Adar Rishon and Adar Sheini are:
"Keriyas haMegillah" - which applies in Adar Sheini alone, and not in Adar Rishon.
"U'matanos la'evyonim" - these too are given to the poor in Adar Sheini and not in Adar Rishon.
The prohibition against delivering eulogies and fasting, however, applies also on the fourteenth and fifteenth of Adar Rishon, even though it is known in advance that this Adar is the first one and that a second Adar will follow it, which is why the Megillah is not read in it.
The reason the Sages established the celebration of Purim specifically in Adar Sheini, and not in Adar Rishon, is in order to place the redemption of Purim adjacent to the redemption that took place at the time of Pesach.
In summary: One who read the Megillah in an Adar during which the year was declared a leap year reads it again in Adar Sheini. The only difference between the two months of Adar is the reading of the Megillah and gifts to the poor, which apply in Adar Sheini alone, whereas the prohibition against eulogies and fasting applies on the fourteenth and fifteenth of Adar Rishon as well. And the reason Purim was established in Adar Sheini is in order to place one redemption adjacent to the other.