The Fast of Gedaliah is observed to commemorate the murder of Gedaliah the son of Achikam, which is described in the last chapter of the Second Book of Kings. This murder resulted in the exile of the Jews who remained in Judea after the Babylonian conquest.
After the first Holy Temple was destroyed (586 BCE) and the Babylonians had exiled the majority of the Jewish people, a small minority were permitted to remain in the Land of Israel. Also, Jews who had fled during the war returned and began to work the land.
Nebuchadnezzar, the Babylonian King, appointed Gedaliah to be the Jewish governor over the remaining population. The King of the neighboring country Ammon, who was vying with the Babylonians for control of the Land of Israel, commissioned Yishmael the son of Netanyah to remove Gedaliah.
Yishmael, who was a descendant of King David, came to the town of Mitzpeh and murdered Gedaliah and all those who were with him. Fearing retribution for the murder of the appointed governor, the remaining Jews fled the Land of Israel, thus completing the exile.
The Fast of Gedaliah is observed on the third day of Tishrei, the day after Rosh Hashana. If the third of Tishrei is Shabbat, the fast is observed on Sunday. The fast begins at dawn and ends at nightfall.*
*Some people rise before dawn to have an early morning breakfast (but this is only permitted if a decision to do so was verbally expressed the night before).