The day after Rosh Hashanah, which is the 3rd of Tishrei, is a public fast day [called Tzom Gedalia]. One is required to fast on this day due to the tragedies that occurred during this time. [This fast is mentioned explicitly in scripture [Zecharia 8] as the “Seventh fast”. The seventh fast refers to the fast of the 3rd of Tishrei which is in the seventh month.]
What occurred on the 3rd of Tishrei: On the 3rd of Tishrei Gedalia Ben Achikam, which was the general and head of the Jewish people after the Churban, was assassinated. This led to the final stage of the destruction, and dispersion and murder of our people.
Announcing the fast the Shabbos before: It is not the Ashkenazi custom to announce the fast of Gedalia the Shabbos before the fast.
When was Gedalia murdered?
It is implied from the classical sources that Gedalia passed away on the 3rd of Tishrei. However some Rishonim learn that he was actually assassinated on Rosh Hashanah, however due to the holiday the fast was delayed until the 3rd of the month, which is the day following R”H.
The purpose of a fast day:
The purpose of a fast day is to arouse the hearts in true repentance after contemplating the suffering we experienced due to our evil ways. Thus each person is to do an accounting of his soul during this time.
One must be especially careful to avoid anger on a fast day.
 602/1; 549/1; Rama 602/1
 Leniencies practiced on Tzom Gedalia: Various sources record leniencies regarding the fast of Gedalia. See Otzer Minhagei Chabad p. 164 that there are many leniencies associated with Tzom Gedalia that came from the desk of the Rebbe Rashab and Rebbe Rayatz. See various sources from Rebbe’s of Poland in Otzer ibid that stated leniencies regarding this fast.
 Rambam Taanis 5/4; See Kaf Hachaim 549/1
 R”H 18b; Rambam Taanis 5/2; See Kaf Hachaim 549/4; M”B 549/2
 Rama 550/4
The Michaber ibid rules that the Shabbos prior to a fast the Chazan should announce the coming fast, with exception to the fasts of Yom Kippur, Tishe Beav and the fast of Esther. The Rama writes that the Ashkenazi custom is to not announce any fast.
 See Taz 549; Kaf Hachaim 549/5
 Rabbeinu Yerucham; Radak brought in Taz 549; Kaf Hachaim ibid
 Rambam Taanis 5/4; Kaf Hachaim 449/2
 Shlah p. 200b; Elya Raba 568/18; Kaf Hachaim 550/11