Shiv'ah 'Asar beTamuz - The Fast of the 17th day of Tamuz - A date when many bad things have happened in Jewish history, most notably the breaching of the walls of Jerusalem by Nebuchadnezzar (586 BCE)
(observed this year on 24 July 2016*)
Tish'ah beAv - the Fast of the 9th day of Av - A date when even more calamitous things have happened in Jewish history, most notably the destructions of both the First and Second Temples in Jerusalem (586 BCE and 70 CE), as well as the expulsion of the Jews from Spain (1492 CE)
(observed this year on 14 August 2016*)
* When the calendar dates of these fasts fall on Shabbat, they are pushed to the next day, meaning the fasts are observed on the Sundays.
The period between these two dates is referred to as The Three Weeks. It is a time of mourning; no parties or weddings are scheduled, and it is customary not to get haircuts.
The last portion of The Three Weeks is the 1st through the 9th of Av (referred to as The Nine Days). The mourning is increased - it is customary not to eat/drink meat and wine, except on Shabbat, and we cancel dancing.