The prophets established a fast day on the tenth of Tevet, Asara Betevet, because that is when Nebuchadnezzar came with his army to besiege Jerusalem, marking the beginning of the series of catastrophes that culminated in the destruction of the First Temple and the exile of the Divine Presence. Though it is true that during the Second Temple period the siege began on a different day, the beginning of the destruction of the Temple and abolition of Jewish dominion occurred on the tenth of Tevet.
Once Asara Betevet was already established as a fast day, the Sages added two other sorrowful events, which occurred around that date, to be commemorated on that day: the death of Ezra the Scribe, which occurred on the ninth of Tevet, and the translation of the Torah into Greek, which occurred on the eighth of Tevet.
The Chief Rabbinate of Israel also expanded the significance of the day, establishing it as a day of general mourning and recitation of Kaddish (Yom HaKaddish Haklali) for the souls of the martyrs who were killed in the Holocaust and whose date of death is unknown.
The Sages say that Ezra the Scribe was worthy enough for the Torah to have been given through him, had Moshe not preceded him (Tractate San. 21b), implying that he was second in stature only to Moses. Ezra enacted ten fundamental decrees (Tractate bk 82a), and by doing so, he inaugurated the rabbinic tradition of the Oral Torah. The Sages further state that Ezra, who ascended from Babylonia to build the Second Temple, was actually the prophet Malakhi (Tractate Megilla 15a). That is to say that, on the one hand, he was the last of the prophets, who are associated with the Written Torah; while on the other hand, he was the first Sage of the Oral Torah. Thus, he was a great Torah scholar who served as a transitional link between the Written Torah and Oral Torah. And just like Moses, he cared for the people of Israel and bore the burden of leading them. He was one of the leaders of the return to the Land of Israel from Babylon and one of the builders of the Second Temple.
Later on, when the Greeks came to power, they issued a harsh decree against the Jewish people, forcing them to translate the Torah into Greek. That day was as calamitous for the Jews as the day on which the Golden Calf was made in the desert, because the Torah belongs to the Jewish people and translating it into Greek blurred its uniqueness, giving the impression that anyone can access it. This occurred on the eighth of Tevet, causing darkness to descend upon the world for three days. Therefore, on Asara Be-Tevet we mention this painful event, as well.
My teacher and mentor, Rabbi Zvi Yehuda Kook, zts"l, said that we must rectify these three issues on Asara Be-Tevet:
1) in response to the siege of Jerusalem, we must strengthen the walls of the city and build up the Land of Israel both spiritually and physically;
2) in response to the death of Ezra the Scribe, we must enhance and glorify the Torah, while working to bring about the ingathering of the exiles, as Ezra did;
3) in response to the translation of the Torah into Greek, we must restore our genuine Jewish spirit and culture and uproot all the evil “spirits” that entered our culture throughout the period of exile, when other nations ruled over us.
Taken from Rabbi Melamed's series on Jewish law "Peninei Halakha:Z'manim". Other interesting and informative articles can be found at:http://en.yhb.org.il/