The Tenth of Tevet (Asara B'Tevet) is a fast day which commemorates the beginning of the seige of Jerusalem by Nebuchadnezzar's armies, which eventually led to the First Temple's destruction and the Babylonian Exile. A number of other tragedies occurred in Jewish history on nearby dates.
The Tenth of Tevet will next be observed on |4+10|, from dawn until nightfall. (See the Aish calendar for fasting times in your location.)