On this date a siege was laid on the city of Yerushalayim. It eventually led to the destruction of the Beis Hamikdash (temple). This is the first in a sequence of four fasts associated with the destruction. In some ways (as the first) it is the most severe.
On this fast day we mourn three separate tragic events: 1. The death of Ezra, 2. the Targum 70 3. the siege of Yerushalayim.
The 8th of TEVES is the date of "Targum Shivim" (Septuagint), the 9th is Ezra's yahrtzeit, The tenth is the day the siege was laid on Yerushalayim. This fast would be fasted on Shabbos (like Yom Kippur). It is in a way more severe than 9 Menachem Av.
The fast of 10 TEVES is compared to Yom Kippur. We would (according to the Abudraham) fast on Shabbos. Why? The Rebbe's talk from 5745 explains this.
On this date in 5738 (1978), the Rebbe reintroduced the age old custom of "Divrei Kivushin", suggesting that words of relevant mussar be spoken in an appropriate way to the community. The Rebbe did so himself and would continue doing so for years to come.
10 Teves is still in the initial stages of Golus and can therefore still be fixed.
Slichos for Asarah BiTeves. inside.
The order of all the Slichos are the same, ony certainly cetain paragraphs are different.
We learnd some of those.
The three reasons for the fats are 1) Torah, 2) Tidden, 3) Eretz Yisroel.
In each the fast is ירידה צורך עלי'ה.