10th of Teves:
The 10th of Teves is a public fast day. 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 “Tenth fast”. The Tenth fast refers to the fast of the 10th of Teves which is in the Tenth month.]
What occurred on the 10th of Teves: On the 10th of Teves the city of Jerusalem was placed under siege by Nebuchadnezzar and from this began the start of the destruction.
Announcing the fast the Shabbos before: It is not the Ashkenazi custom to announce the fast of Teves the Shabbos before the fast.
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.
 Rambam Taanis 5/4; See Kaf Hachaim 549/1
 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.
 Rambam Taanis 5/4; Kaf Hachaim 449/2
 Shlah p. 200b; Elya Raba 568/18; Kaf Hachaim 550/11