Shavuot, the Festival of Weeks, is the second of the three major festivals with both historical and agricultural significance (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 Hag ha-Bikkurim (the Festival of the First Fruits). Historically, it celebrates the giving of the Torah at Mount Sinai, and is also known as Hag Matan Torateinu (the Festival of the Giving of Our Torah).
The period from Passover to Shavuot is a time of great anticipation. We count each of the days from the second day of Passover to the day before Shavuot, 49 days or 7 full weeks, hence the name of the festival. See The Counting of the Omer. Shavuot is also sometimes known as Pentecost, because it falls on the 50th day. The counting reminds us of the important connection between Passover and Shavuot: Passover freed us physically from bondage, but the giving of the Torah on Shavuot redeemed us spiritually from our bondage to idolatry and immorality.
It is noteworthy that the holiday is called the time of the giving of the Torah, rather than the time of the receiving of the Torah. The sages point out that we are constantly in the process of receiving the Torah, that we receive it every day, but it was first given at this time. Thus it is the giving, not the receiving, that makes this holiday significant.
Shavuot is not tied to a particular calendar date, but to a counting from Passover. Because the length of the months used to be variable, determined by observation (see Jewish Calendar), and there are two new moons between Passover and Shavuot, Shavuot could occur on the 5th or 6th of Sivan. However, now that we have a mathematically determined calendar, and the months between Passover and Shavuot do not change length on the mathematical calendar, Shavuot is always on the 6th of Sivan (the 6th and 7th outside of Israel. See Extra Day of Holidays.)
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. See Separation of Meat and Dairy.
The book of Ruth is read at this time. Again, there are varying reasons given for this custom, and none seems to be definitive.
Shavuot will occur on the following days on the American calendar:
Sources: Judaism 101