On the tenth of Teves in the ninth year of the reign of Tzidkiyahu, the last king of Yehudah, Nevuchadnezzar, King of Bavel, laid siege to Yerushalayim. The siege lasted until the eleventh year of Tzidkiyahu's reign, when, on the seventeenth of Tammuz, the walls of Yerushalayim were breached, leading to the destruction of the Beis Hamikdash on the ninth of Av (Tisha B'Av).
The prophet Yechezkel was in Bavel at the time of the start of the siege, and yet he had a prophesy on that tenth of Teves, informing him that the siege had begun.
"And the word of Hashem was unto me in the ninth year, in the tenth month, on the tenth of the month, saying: "Son of man, write for yourself the name of this day, this very day, the king of Bavel has laid siege to Yerusholayim on this very day."
Because the destruction of the Beis Hamikdosh and all the great calamities that came in its wake began with this siege, the fast of Asarah B'Teves was instituted for all future generations.
So significant is this fast that Avudraham (quoted by Beis Yosef, Orach Chaim, 550) writes that if it were ever to fall on a Shabbos, we would have to fast even though it is Shabbos. This, he notes, is implied by the prophet emphasizing "on this very day," to teach that one must fast even if it falls on Shabbos. In fact, according to the fixed calendar that we have today, Asarah B'Teves can never fall on a Shabbos, although it is the only fast that can sometimes, as this year, be observed on Erev Shabbos.
