Asara B’Tevet

Tenth of Tevet (Hebrew: עשרה בטבת‎, Asarah BeTevet), the tenth day of the Hebrew month of Tevet, is a fast day in Judaism. It is one of the minor fasts observed from before dawn to nightfall. The fasting is in mourning of 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 (the First Temple) and conquest of the Kingdom of Judah.
The fast day is not related to Hanukkah but happens to follow that festival by a week. Whether the 10th of Tevet occurs 7 or 8 days after the last day of Hanukkah depends on whether the preceding Hebrew month of Kislev has 29 or 30 days in the relevant year.