This coming Sunday, December 23rd 2012, we observe the fast of the Tenth of Tebet. We remember the siege of Yerushalaim, in the year 586 BCE, at the time of the destruction of our first Bet haMiqdash (see this).
On the 10th of Tebet there are only two prohibitions: eating and drinking. The fast begins at dawn and it ends at 5:03 p.m. NYT.
NO additional limitations apply, such as the prohibition of wearing leather shoes, working, driving, washing the body, etc.
Most contemporary Rabbis (R. E. Melamed, Rab O. Yosef) authorize to wash one's mouth or brush one's teeth in this Ta'anit, when necessary, provided you are very careful to lower your head, avoiding swallowing water unintentionally.
In modern Israel, the 10th of Tebet is also recognized as the day of the Kaddish haKelaly. According to the Chief Rabbinate of Israel, on the Tenth of Tebet a remembrance-candle should be lit in the Synagogue and the Hazkara leHalale haShoah should be recited. Additionally, all those whose parents are not alive should say the Kaddish Yatom (luach dinim uminhaguim 5772, pages. 55,109).
This point requires more explanation.
In 1949, and before the day of Yom HaShoah was established, the Chief Rabbinate of Israel decided that the Tenth of Tebet should be assigned as the national remembrance day for the victims of the Holocaust. They recommended traditional Jewish ways of remembering the dead, such as the study of Mishna Mikvaot, saying Tehilim, lighting a candle and a communal recitation of the Kaddish for the victims of the Holocaust whose names and date of death remain unknown. Fasting, the most common Jewish expression of sorrow, was already prescribed for this day.
In Israel many people felt that the horror of the Holocaust should be remembered on its own, and a special day should be dedicated to the Shoah's victims' memory. "For the Holocaust survivors there was only one day worthy of being a memorial anniversary for the Holocaust--April 19, the beginning day of the Warsaw ghetto revolt the greatest revolt of them all, the uprisings that had held the Nazis at bay for a longer period than the great French army" (I. Greenberg). That is how the 27 of Nissan was chosen to commemorate Yom haShoah. Yom HaShoah was inaugurated in 1953, by a law signed by the Prime Minister of Israel David ben Gurion.
Since then, and in practical terms, there are two days in which we mourn for the Holocaust: Yom haShoah, the official day, and'asara beTebet, in which people say the Kaddish haKelaly to remember the victims of the Nazi genocide.
Candle lighting in NYC: 4:14 p.m.
Shabbat ends in NYC: 5:12 p.m.
Who is exempted from fasting on the Tenth of Tebet?
*Minors: boys under 13 and girls under 12 years old are completely exempted from fasting.
*Nursing women: According to the Sephardic Minhag, after giving birth women are exempted from fasting for 24 months, even if they are not actually nursing their baby.
*Pregnant women, especially after the first 3 months, are exempted from fasting.
*A person who feels sick--for example, flu or fever-- or one who has a chronic disease--for example diabetes-- should not fast.
*Elders should consult with their physicians if the fast will not affect their health. If it will, they are exempted (and in some cases, prohibited) from fasting.