Dreidel is a four-sided top used to play a children's gambling game on Chanucah. Each side is inscribed with one of the letters nun, heh, gimmel, shin, standing for nes gadol hayah sham, "there was a great miracle there". In Israel shin for sham is replaced by peh for poh, meaning here - the miracle was "here".
This is how you play. Every participant gets a bunch of "tokens", such as chocolate money and then spins the dreidel in turn.
Each letter stands for an outcome; eg gimmel is short for ganz in Yiddish, meaning that you take everything in the pot etc. At the end of the game you eat the money.
The most common connection to Chanucah that I've heard is that at the time of Antiochus, the evil Greek ruler, Jews were forbidden to learn Torah. They would gather secretly to do so anyway and, if discovered, would whip out their dreidels and pretend to be playing a harmless children's game.
Dreidel has come far since then and one can now play the game at a serious competitive level. The Major Dreidel League was founded in New York in 2007 and holds Chanucah tournaments in its stadium, the spinagogue. A number of commercially produced dreidel games are available in the US, including Stacabees and No Limits Texas Dreidel.