Tzom (the fast of) Gedalyah (or Gedaliah) is a minor fast day which takes place every year on the 3rd of Tishrei (the day immediately after Rosh HaShanah), unless that day happens to be a Shabbat, in which case the fast is postponed to the following day (this is because Shabbat is considered to be a happy day and the rabbis therefore consider it inappropriate to fast on Shabbat except in the case of Yom Kippur).
Who was Gedalyah?
In 586BCE the Babylonians conquered Jerusalem and, following their usual practice, exiled many (not all) of its former inhabitants (they were forcibly taken to Babylon). The Babylonians appointed Gedalyah, a Jew, to govern the province of Judah and its capital Jerusalem on their behalf. Of course, Gedalyah was expected to implement the policies and wishes of the Babylonians. However, Gedalyah did also where possible use his position to moderate how these policies were implemented.
However, Gedalyah was assassinated in Tishrei about 3 years after his appointment as governor. The assassination was instigated by the ruler of the neighbouring kingdom of Ammon who persuaded the group of Jews who carried out the assassination that Gedalyah was a collaborator. Details of the assassination can be found in the Tanach (Jewish Bible) at II Kings 25: 25-26 and in Jeremiah chapter 41.
Angered by the assassination, the Babylonians then appointed a non-Jewish governor who ruled very harshly and unsympathetically. Consequently, life for the remaining Jewish inhabitants became even harder than it had been previously.
The sages of the period viewed the assassination of Gedalyah as causing the loss of the last vestiges of Jewish influence in the governing of Judah and Jerusalem. Although there is some uncertainty about the exact date on which Gedalyah was assassinated, the sages ruled that the 3rd of Tishrei each year should be observed as a fast day to commemorate and mourn the loss of Jewish autonomy under the Babylonians.