We fast on the Tenth of Tevet, which occurs this year on Tuesday December 30th to commemorate the beginning of the siege of Jerusalem by Nebuchadnezzar in 588 BCE.  The siege ultimately resulted in the sacking of Jerusalem and the destruction of the first temple, so we note that this is when that whole tragedy began.  

It is a minor fast, so we observe fasting from sunup (6:18arn this year) until sundown (5:38pm).

There are numerous other events  associated with the day:  the Arba Turim mentions that in the Second Temple times, King Ptolemy of Egypt secluded 70 of our great rabbonim in separate chambers, and ordered them to translate the Torah into Greek.  He expected that since there such a divergence of translation possibilities, the results would give skeptics an opportunity to make light Of sages' wisdom. By divine inspiration, each of the seventy chose exactly the same words to translate ambiguous passages; the result is

known today the Septuagint, or "translation by the seventy." 

Additionally, this day has been designated by the Israeli Chief Rabbinate as a universal day of kaddish for Holocaust martyrs. Many people who lost family in the Holocaust do not know the date on which to properly commemorate the yahrtzeit; so the 10th of Tevet is designated as the day to say kaddish for those who died on an unknown date.