By chabad.org
On the 10th of Tevet of the year 3336 from Creation (425 BCE), the armies of the Babylonian emperor Nebuchadnezzar laid siege to Jerusalem.
Thirty months later — on Tammuz 9, 3338 — the city walls were breached, and on Av 9th of that year, the Holy Temple was destroyed. The Jewish people were exiled to Babylonia for 70 years.
Tevet 10 is observed as a day of fasting, mourning and repentance. We refrain from food and drink from daybreak to nightfall, and add the Selichot and other special supplements to our prayers.
More recently, Tevet 10 was chosen to also serve as a “general kaddish day” for the victims of the Holocaust, many of whom the day of their martyrdom is unknown.
An ancient Jewish custom, which was revived by the Lubavitcher Rebbe, is to deliver words of inspiration and arousal to repentance on fast days.
BTW there is a clip going around of an audio of the Rebbe saying Slichos (and crying) on Asara B’Teves תשכ”ה . The video is showing the Rebbe davening by the Amud in the downstairs Shul in 770. However the Selichos was actually said by the Rebbe then in the small upstairs Zal in 770