The Tamid (daily) sacrifice was discontinued on this day shortly before the destruction of the first Beit HaMikdash (the Holy Temple) because Jerusalem was under a state of siege and they were unable to get the necessary sheep.
Jerusalem's city walls were penetrated by the Romans before the destruction of the second Beit HaMikdash (and possibly by the first as well).
An idol was erected in the Beit HaMikdash on this day. (There is a difference of opinion in the Talmud (Yerushalmi Taanis 4:5) whether this is talking about the first or second Beit HaMikdash.)
The Torah was burnt on this day by Apustemus, one of the Greek oppressors.
In memory of these events we are required to fast on this day to inspire ourselves to repentance. The fast begins at the break of dawn and ends at nightfall.
During this time we neither eat nor drink any food whatsoever, not even water. Even though we are, strictly speaking, permitted to bathe on this fast day (unlike Tisha BeAv and Yom Kippur) the custom is not to bathe on Shiva Asar BeTamuz.
Pregnant or nursing women, as well as anyone else for whom fasting may be a health problem should consult with a rabbi whether they are permitted to fast. Children below the age of majority (bar or bas mitzvah - 13 for boys and 12 for girls) do not fast. (In some communities, it is customary for children to begin fasting a short time before they become bar/bas mitzvah.)
It is important to recognize that the primary idea behind a fast is to meditate on the fact that these sufferings came upon us because the sins of our ancestors, sin which we continue to commit, and that we must repent. Someone who fasts but spends the day in frivolous activity has completely missed the point.
The fast of Shiva Asar B'Tamuz marks the beginning of a three week period of national mourning for the Jews which is completed on Tisha BeAv, the ninth of Av.