Shavuot, Hebrew for "weeks", is so named because it occurs seven weeks after Pesach (Passover). This period of seven weeks is called the omer. The omer is treated like a semi-mourning period, except for the 33rd day, Lag BaOmer, which falls on the 18th day of the Jewish month of Iyar and is a festive day for which the mourning behavior is suspended. Lag BaOmer is typically celebrated with a picnic during the day and/or a bonfire at night, and Jewish schools often hold Color Wars or Track and Field Day for their students.
Shavuot itself is a celebration of the anniversary of the Jewish people's receiving the Torah at Mount Sinai. It is customary to gather together and stay up all night studying Torah and other Jewish texts the first night of Shavuot, in sessions called Tikkun Leil Shavuot. The book of Ruth is read during synagogue services because her conversion to and acceptance of Judaism is analogous to the Jewish people's accepting of the Torah (and also because the story takes place around the same time of year as Shavuot). Some religious school graduations or confirmations are also held on Shavuot.
While most Jewish holidays are celebrated by eating meat or fish, on Shavuot it is customary to eat dairy items like cheesecake, blintzes, or bourekas. One explanation is that when the Torah was first handed down, the Jews realized that the meat they were eating wasn't kosher, so they ate dairy items instead. Others say that it is a reference to the Biblical phrase referring to Israel as "the land of milk and honey." It could also be because cheese is produced during the spring harvest season, which Shavuot also marked in ancient times (the omer spanned seven weeks between the first barley harvest and the first wheat harvest, after which offerings of both, along with ripened fruits, were brought to the Holy Temple).
The Learning Session schedule for JFGI's Tikkun Leil Shavuot event (see below) is now live!