The Talmud (Taanit 28b) lists five tragic events in Jewish history that occurred on Tammuz 17, on account of which a fast was instituted on this day (see Laws & Customs").
The first of these occurred in 1313 BCE, forty days after the Giving of the Torah on Sivan 6. Upon descending Mount Sinai and witnessing Israel's worship of the Golden Calf (see "Today in Jewish History" for yesterday, Tammuz 16), Moses smashed the Tablets inscribed with the Ten Commandments which he was carrying down from the mountain.
(for the other four tragedies of Tammuz 17, see below)
Links:
Broken & Whole
The 120-Day Version of the Human Story
Moses Breaks the Tablets
The daily sacrificial offerings (Korban Tamid) in the Holy Temple were discontinued, three weeks before the Babylonians' destruction of the First Temple in 423 BCE.
The other three national tragedies mourned on Tammuz 17 are connected with the Roman conquest of Jerusalem and their destruction of the Second Temple in the year 69 CE:
--The walls of the besieged city of Jerusalem were breached.
--The Roman general Apostomus burned the Torah and,
--placed an idol in the Holy Temple.
The fighting in Jerusalem continued for three weeks until the 9th of Av, when the Holy Temple was set aflame.
Tammuz 17 is a fast day, devoted to mourning the breaching of Jerusalem's walls and the other tragic events that occurred on this day (see "Today in Jewish History") and repenting and rectifying their causes. We refrain from all food and drink from "daybreak" (about an hour before sunrise, depending on location) until nightfall. Special prayers and Torah readings are added to the day's services.
Link: Halachic times for Tammuz 17
The 17th of Tammuz also marks the beginning of The Three Weeks period of mourning which culminates on the 9th of Av, commemorating the conquest of Jerusalem, the destruction of the Holy Temple and the dispersion of the Jewish people.
Weddings and other joyful events are not held during this period; like mourners, we do not cut our hair, and various pleasurable activities are limited or proscribed. (Consult the Code of Jewish Law (Shulchan Aruch) or a qualified rabbi regarding specific proscriptions).
The Lubavitcher Rebbe urged that the Three Weeks should be a time of increased giving of charity and Torah study (in keeping with the verse (Isaiah 1:27), "Zion shall be redeemed by law, and her returnees by charity"), particularly the study of those portions of Torah that deal with the laws and the deeper significance of the Holy Temple.
Links:
TheThreeWeeks.com
Some Laws and Customs of the Three Weeks
About Holy Temple