The Tenth of Tevet (Heb, Asara b'Tevet) marks the day on which the Babylonian siege of Jerusalem began in the year 588 BCE, an event which eventually led to the destruction on the Temple in 586 BCE and the first exile from Israel. Though the day usually falls out near the time of Hanukkah, the two holidays have no significant relationship with each other. The Tenth of Tevet is considered a "minor fast" and orthodox Jews refrain from eating or drinking from sunrise to sunset on the day of fasting.
In Israel, the Tenth of Tevet has also come to be marked as a memorial day for the 6 million Jewish victims of the Holocaust. The kaddish (Prayer for the deceased) is recited on this day for people whose date or place of death during the Holocaust is unknown and that is how the date has gotten the unofficial Hebrew moniker, Yom Ha-kaddish ha-kleli, literally translated as The General Kaddish Day.
And it was in the ninth year of his reign, in the tenth month, in the tenth (day) of the month, that Nebuchadnetzar, King of Babylon came, he and all his hosts, upon Yerushalayim, and he encamped upon it and built forts around it. And the city came under siege till the eleventh year of King Tzidkiyahu. On the ninth of the month famine was intense in the city, the people bad no bread, and the city was breached (Second Melachim 25).
We see then, that the tenth of Tevet on which the siege of Yerushalayim began, was the beginning of the whole chain of calamities which finally ended with the destruction of the Beit Hamikdash.
'The essential significance of the fast of the Tenth of Tevet, as well as that of the other fast days, is not primarily the grief and mourning which they evoke. Their aim is rather to awaken the hearts towards repentance; to recall to us, both the evil deeds of our fathers, and our own evil deeds, which caused anguish to befall both them and us and thereby to cause us to return towards the good. As it is said: And they shall confess their transgressions and the transgressions of their fathers. (Vayikra 26. Rambam Hilchot Ta'anit Chapter 5). The aim of fasting, therefore, is to subjugate our evil inclination by restriction of pleasure; to open our hearts and stir us to repentance and good deeds through which the gates of Divine mercy might be opened for us.
Therefore, each person is obligated to examine his deeds and to repent during these days. As it is written of the people of Ninveh: And the Lord saw their actions (Yonah 3) . . upon which the Rabbis say: It is not said, He saw their sackcloth and fasting, but rather their actions (Ta'anit 22). We see hence that the purpose of fasting is repentance.
'Therefore, the people, who fast but engage in pointless activities, grasp what is of secondary importance and miss what is essential. Nevertheless, repentance alone without fasting is also insufficient. There is a positive commandment of Rabbinic origin to fast on his day.
If a public fast falls on Shabbat, it is delayed until after Shabbat since fasting is not permitted on Shabbat. The one exception is Yom Kippur, which based on a verse in the Torah is observed even if it falls on Shabbat. The Geonim also write that the same was once true of the tenth of Tevet, since it is written of the tenth of Tevet: On this very day (YehezkeI 2). In our calendar calculation, however, the tenth of Tevet can never fall on Shabbat.
It a public fast occurs on Erev Shabbat we fast the entire day till the conclusion of the fast, even though it means entering Shabbat while fasting. Nowadays our calendar calculation is such that the only public fast, which can fall on Erev Shabat, is the tenth of Tevet.
Sources: Orthodox Union