The Fast of Gedalia (Or Tzom Gedaliah in Hebrew) is observed on the 3d day of the Hebrew month "Tishrei", immediately after Rosh Hashanah. It is also called "Tsom Ha'Shviii", meaning "The fast of the 7th month". The fast begins in the morning of the day following Rosh Hashanah, and ends at sunset.
The origin of this fast is explained in two places in the bible: Kings II, chapter 25; The Book of Jeremiah, chapters 40-41. It is written that after the destruction of the 1st Temple by the Babylonians (in the year 582 BC), few Jews remained in Judea, mainly poor vine-growers. Nebuchadnezzar, the king of Babylon, appointed a man named Gedaliah son of Ahikam to be the king of Judea. Gedaliah settled in a place called Mizpah, together with his followers and the prophet Jeremiah. During his attempts to revive the Jewish autonomy, a man called Yishmael, who was a descendant of the royal family, arrived with ten other men and murdered Gedaliah along with many others. The remaining people fled to Egypt, and that was the end of the Jewish settlement in Israel for many years.
This murder could be explained in several ways: It could have been personal hatred, along with jealousy and objection to having a king who was not a descendant of the royal family. It could also be seen as a political murder, for Gedaliah was willing to operate under Babylonian rule. An additional explanation could be that Yishmael was influenced by the king of Amon, who led him to this murder, hoping to create chaos in Judea (which would allow him to invade and expand the borders of his own state).
In any case, this murder was far more than "just" a murder: It was a political murder, commited by a fellow Jew, and it so happened that this terrible sin was done during "The Days of Awe", between Rosh Hashanah and Yom Kippur. As to the symbolism of this murder, it signified the end of Jewish Monarchy, and led to the destruction of what remained of the Jewish settlement in Israel. This all happened soon after the Temple was destroyed. It strengthened the sense of despair, and therefore the day of the murder was declared a day of mourning.