This fast day comes less than two weeks after Chanukah ends, just when you thought there were no more fast days for a while. It marks the day the siege of Jerusalem began in 586 B.C.E. Like all fast days, it's a call to action for repentance.
In Israel, Asarah B'Tevet also marks the day Kaddish, the Jewish memorial prayer, is recited for people whose date or place of death is unknown. Naturally, it's therefore thought of as a day of mourning for many who perished in the Holocaust.