There are four minor fasts on the Jewish calendar. These fasts were instituted by the Sages to commemorate some national tragedy. The minor fasts (that is, all fasts except Yom Kippur and Tisha b'Av) last from dawn to nightfall, and one is permitted to eat breakfast if one arises before sunrise for the purpose of doing so. There is a great deal of leniency in the minor fasts for people who have medical conditions or other difficulties fasting. The date of the fast is moved to Sunday if the specified date falls on a Sabbath.
Three of these four fasts commemorate events leading to the downfall of the first commonwealth and the destruction of the first Temple, which is commemorated by the major fast of Tisha B'Av.
Following is a list of minor fasts required by Jewish law, their dates, and the events they commemorate:
The Fast of Esther, Adar 13, commemorates the three days that Esther fasted before approaching King Ahasuerus on behalf of the Jewish people. The fast is connected with Purim. If Adar 13 falls on a Friday or Saturday, it is moved to the preceding Thursday, because it cannot be moved forward a day (it would fall on Purim).
Source: Judaism 101