Tenth of Tevet

Tenth of Tevet (Hebrew: עשרה בטבת), the tenth day of the Hebrew month of Tevet, is a minor fast day in Judaism. It is a observed from sunrise to sunset. The day has no relationship to the Jewish Holiday Hanukkah, but it happens to follow that festival by a week. Whether the 10th of Tevet falls seven or eight days after Hanukkah depends on whether the preceding month of Kislev has 29 or 30 days in the relevant year.

According to Israel History, the fast commemorates the siege of Jerusalem by Nebuchadnezzar II of Babylonia – an event that began on that date and ultimately culminated in the destruction of Solomon’s Temple and the conquest of the Judah.
As with all minor Jewish fast days, the Tenth of Tevet begins at dawn and concludes at nightfall in accordance with the general rules of minor fasts as set forth in the Code of Jewish Law, and in contrast to Tisha B’Av, there are no additional physical constraints beyond fasting.
A Torah reading, a special prayer in the Amidah (Aneinu), and  the Avinu Malkeinu prayer is added at both Shacharis and Mincha services. At Shachari’s services, the Selichos are also said, and at Mincha, in Ashkenazic congregations, the Haftarah is read.
If a public fast occurs on Erev Shabbat, we fast the entire day until the conclusion of the fast, even though it means entering Shabbat while fasting. Neither nor “Tachanun” is recited at Mincha.  The fast continues until after the completion of Maariv for Shabbat.  One should not eat or drink anything until after Kiddush.
The Chief Rabbinate of Israel chose to observe the Tenth of Teves as a kaddish for the victims of the Holocaust.

Credits:
https://en.wikipedia.org/wiki/Tenth_of_Tevet