Shavuot is the festival which celebrates the giving of the Torah, the five books of Moses, to the Israelites at Mount Sinai. When they arrived at the mountain, it was cordened off and anybody who passed the barrier was immediately put to death. Only Moses was allowed to cross, and he went up and received the Ten Commandments on two stone tablets. However, while he was gone, the Israelites feared for his life, and started to disbelieve in G-d, so they commanded Aaron to build them a Golden Calf. When moses came down from the mountain and saw the calf, he smashed the two stone tablets and smashed the calf. He then returned up the mountain to receive another two tablets.
On Shavuot, we recite the Hallel and read from the Book of Ruth. Shavuot is a festival of learning - tradition dictates that people stay up all night studying Torah.
In relation to other festivals, Shavuot lies seven weeks after Pesach, the period between the festivals being called the Omer.
Over the two day festival, we eat cheesecake and blintzes, because the Israelites did not want to spend their time around Sinai preparing a meat meal, and so ate dairy products.