Rosh Chodesh Tevet is sometimes observed as one day and sometimes as two, because Kislev, the preceding month, is sometimes "full" (consisting of 30 days) and sometimes not (consisting of only 29 days). With a two-day Rosh Chodesh, its first day is the 30th day of the preceding month, and its second day is the 1st day of the next month. The month of Tevet itself always consists of precisely 29 days; because of this lack of variation in the length of Tevet, the Rosh Chodesh of Shevat, the month which follows, always consists of only one day, namely, the 1st of Shevat.
During the month of Tevet, three fast days are observed, the 8th, 9th and 10th of the month, in commemoration of three major calamities which befell the People of Israel.
The fast days of the 8th and 9th of Tevet are called 'fast-days-for-the-righteous,' as on these days, only individuals who choose to, fast, whereas the fast of the 10th of Tevet is a public fast obligating the entire Jewish community.
0n the 8th of Tevet, at the beginning of the "Greek Era," the Torah was translated into Greek by the decree of King Ptolemy, in about the year 313 B.C.E., according to the Bayit Sheni Timeline presented in the Chanukah Section, which locates the date of the Destruction of the First Temple in 423 B.C.E. This corresponds to the approximate year of 476 B.C.E., according to the alternate timeline which locates the Destruction at 586 B.C.E. That day was regarded as equally calamitous for Israel as the day on which the Golden Calf was made, since it is impossible to adequately translate the Torah.
Moving backwards in time, the 9th of Tevet commemorates the death of both Ezra the Scribe and Nechemia, two of our greatest leaders, who faithfully led Israel during the return from Babylonian captivity. This occurred in approximately the year 353 B.C.E., according to the first Timeline referenced above, or approx. 516 B.C.E., according to the alternate Timeline. It is said that the eyes of all Israel were clouded by their death, for the loss of these great leaders was a tremendous blow, and Ezra's stature, in particular, was so great that it is said of him that had the Torah not been given to Israel through Moses, it would have been given to Israel through Ezra.
On the 10th of Tevet, Nevuchadnetzar, King of Babylonia, laid siege to Yerushalayim. The imposition of this vise-like siege occurred in the approximate year 426 B.C.E., according to the Bayit Sheni Timeline, or the approximate year 589 B.C.E., according to the timeline which sets the year of the Destruction of the First Temple as 586 B.C.E.
The walls of Yerushalayim were subsequently breached on the 17th of Tammuz, during the third year of the siege, and the Temple destroyed on Tisha B'Av, the 9th of Av, the saddest day in the Hebrew Calendar, the Date of Destruction of both Temples, and the most somber fast day. But the start of the siege set the stage for both of those tragic events. It was for this reason that the Prophets and other Jewish Leaders of the time set the Tenth of Tevet as a day of fasting for the Jewish People.
In modern day Israel, the 10th of Tevet has been designated as Yom Hakaddish Haklali - the day on which we mourn those whose date or place of death is unknown.
On the 1st of Tevet, Yechoniah, King of Yehudah, was exiled together with the Sages and the nobility, members of the family of royalty, of Yerushalayim. The day was not however designated as a public fast day.
With the exception of the closing days of Chanukah, the month of Tevet contains no Yom Tov or festive day.