And from... the day after the sabbath, you shall count off seven weeks. They must be complete: you must count until the day after the seventh week—fifty days... On that same day you shall hold a celebration; it shall be a sacred occasion for you; you shall not work at your occupations. This is a law for all time in all your settlements, throughout the ages.
-Leviticus 23:15-16, 21
Shavuot, the Festival of Weeks, is the second of the three major festivals (the other two are Passover and Sukkot). Agriculturally, it commemorates the time when the first fruits were harvested and brought to the Temple, and is known as Chag ha-Bikkurim (the Festival of the First Fruits). Historically, it celebrates the giving of the Torah at Mount Sinai, and is also known as Z'man Matan Torateinu (the Time of the Giving of Our Torah).
The period between Passover and Shavuot is a time of great anticipation. We count each of the 49 days from the second day of Passover until the day before Shavuot - 7 full weeks (hence the name). This period of counting is called Sefirat Ha-Omer (the Counting of the Omer). The counting reminds us of the important connection between Passover and Shavuot: While Passover freed us physically from bondage, the giving of the Torah on Shavuot redeemed us spiritually from our bondage to idolatry and immorality.
It is customary to stay up the entire first night of Shavuot to study Torah. This is called Tikkun Leil Shavuot. Sinai honors this tradition by holding an annual learning program until past midnight on the first night of Shavuot.
It is customary to eat a dairy meal at least once during Shavuot. There are varying opinions as to why this is done. Some say it is a reminder of the promise regarding the land of Israel, a land flowing with "milk and honey." According to another view, it is because our ancestors had just received the Torah (and the dietary laws therein), and did not have both meat and dairy dishes available. Congregation Sinai holds an annual "Bring Your Own Dairy Lunch Picnic" at the synagogue on the second day of Shavuot.
We also read the book of Ruth on Shavuot.