'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 had 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 (Vayikra 26): 'And they shall confess their transgressions and the transgressions of their fathers.' (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.' (Chayei Adam; Klal 33)
'Therefore, each person is obligated to examine his deeds and to repent during these days. As it is written of the people of Nineveh: '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.' (same as above)
'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, because there is a positive commandment of Rabbinic origin to fast on this day.' (same as above)
The fast begins, as do all the public fasts with the exception of Tisha B'Av and Yom Kippur, at "alot haShachar," "dawn."
These four public fasts: Asarah B'Tevet, Tzom Gedaliah, Shiva Asar B'Tammuz, and Ta'anit Esther, are also alike in that there do not apply any additional physical constraints, such as the prohibition of washing or of wearing leather shoes, etc.
One who is in the category of being ill-even-without-danger, or who is a pregnant or nursing woman, for whom fasting might be difficult, and all children, are exempt from fasting.
All those who are exempt from fasting should not, in any case, eat publicly or indulge in purely pleasurable forms of consumption (such as eating black-'n-whites), but should eat only that which is necessary for good nutrition.
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' (YechezkeI 2). In our calendar calculation, however, the tenth of Tevet can never fall on Shabbat.
If 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. Neither "Avinu Malkeinu" nor "Tachanun" are recited at Mincha. The fast continues until after the completion of Maariv for Shabbat (after the appearance of the stars). One should not eat or drink anything until after Kiddush.
Nowadays our calendar calculation is such that the only public fast which can fall on Erev Shabbat is the tenth of Tevet.