Chanukah is the Festival of Lights, commemorating the rededication of the Temple in Jerusalem after a successful revolt by a band of Jewish fighters called the Maccabees against the Syrian-Greek empire that had attempted to destroy the Jewish faith. As part of the rededication, the victorious Jews needed to light the Temple's menorah (seven-branch candelabrum), but they only had enough oil to last one day, and it would take eight days to prepare more. Miraculously, the one-day supply of oil lasted for eight days, which is why Chanukah lasts for eight days.
A chanukiah (nine-branch menorah — one candle for each day of Chanukah plus another candle, called the shamash, to light them) is lit each night, starting with one candle on the first night and adding a candle each day until the whole chanukiah is lit on the last night of Chanukah. The chanukiah is traditionally placed in a window facing the street, sharing the light and celebration with the world.
It is traditional to eat fried foods on Chanukah to commemorate the miracle of the oil. These include latkes (potato pancakes) and sufganiyot (jelly doughnuts), among others.
Chanukah gifts have traditionally been small — pocket money or something similar (this is where Chanukah gelt, or chocolate coins, comes from). As nearby holiday Christmas grew in influence, so did Chanukah gifts.
Gelt are often used as the prize for winning a game of dreidel, traditionally played on Chanukah. Legend has it that if a Syrian-Greek soldier was approaching a group of Jews studying Torah, they would quickly hide their books and start playing dreidel instead. Each side of the dreidel (spinning top) has a different Hebrew letter on it - Nun (for nes), Gimmel (for gadol), Heh (for hayah) and Shin or Peh (for sham or poh). Nes gadol hayah sham/poh means "A great miracle happened there/here" (dreidels with the latter only exist in Israel). Each letter also indicates a different move that the player should make - nun means "nothing", gimmel means "take the whole pot" (usually chocolate gelt, sometimes nuts or perhaps real money), heh means "take half the pot", and shin or peh means "put a piece of your stash into the pot."
Chanukah sameach (Happy Chanukah)!