There are five minor fasts on the Jewish calendar. With one exception, these fasts were instituted by the Sages to commemorate some national tragedy. Minor fasts (that is, all fasts except Yom Kippur and Tisha b'Av) last from dawn (first light) to nightfall (full dark), and one is permitted to eat breakfast if one arises before dawn for the purpose of doing so (but you must finish eating before first light). 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 Shabbat.
Three of these five fasts commemorate events leading to the downfall of the first commonwealth and the destruction of the 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 Gedaliah, Tishri 3, commemorates the killing of the Jewish governor of Judah, a critical event in the downfall of the first commonwealth.
The Fast of Tevet, Tevet 10, is the beginning of the siege of Jerusalem. It has also been proclaimed a memorial day for the six million Jews who died in the Holocaust.
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 Shabbat, it is moved to the preceding Thursday (Adar 11), because it cannot be moved forward a day (it would fall on Purim).
The Fast of the Firstborn, Nissan 14, is a fast observed only by firstborn males, commemorating the fact that they were saved from the plague of the firstborn in Egypt. It is observed on the day before Passover. If Nissan 14 falls on Shabbat, the fast is moved back to the previous Thursday to avoid interfering with Shabbat. Like the Fast of Esther, it can't be moved forward because it would fall on the holiday.
The Fast of Tammuz, Tammuz 17, is the date when the walls of Jerusalem were breached by the Romans in 69 C.E., a major event leading up to the destruction of the Temple exactly three weeks later, on the 9th of Av. Those three weeks are a time when mourning customs are observed. There are three special haftarah readings of admonition for each of these three weeks.