There are four minor fasts on the Jewish calendar that were instituted by the Sages to commemorate a specific national tragedy. These minor fasts last from before sunrise to nightfall, as opposed to Yom Kippur and Tisha B'Av that last for from sundown to sundown the following day. There is a great deal of leniency in the minor fasts for people who have medical conditions or other difficulties fasting and date of the fast is moved to Sunday if the calendar date falls on a Sabbath.
The following is a list of minor fasts required by Jewish law, their dates, and the events they commemorate:
The Fast of Gedaliah
Commemorates the killing of the Jewish governor of Israel, a critical event in the downfall of the first commonwealth. (Tishrei 3)
Tenth of Tevet
Also known by its Hebrew name, Assara B'Tevet, it marks 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.
Fast of Esther
Commemorates the three days that Esther fasted before approaching King Ahasuerus on behalf of the Jewish people. The fast is connected with Purim and falls on Adar 13. 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).
Seventeenth of Tammuz
Is the date when the walls of Jerusalem were breached.