Hanukkah is the celebration of the cleansing of the Temple in Jerusalem in 168 B.C. It had been taken over by the Greeks and dedicated to their god Zeus. Hanukkah was originally a minor holiday, but its proximity to Christmas is partly why it has grown in significance.
The word Hanukkah means dedication in Hebrew, and the holiday is also known as the Festival of Lights. It is celebrated over eight days, during which Jewish families light one candle on the first night, two on the second and so on until all eight candles are lit on the final night. The holiday occurs on the 25th day of the month Kislev in the Jewish calendar. This is a lunar calendar, which means that it doesn't coincide with the Gregorian calendar, so Hanukkah can be anytime in late November to December.
Hanukkah was originally not a religious holiday, and is the only Jewish holiday that is not mentioned in the Torah. In fact, it wasn't until the 1920s that Jews in the United States started giving more importance to the holiday, after watching their fellow Americans celebrating Christmas. It was at this time that the practice of giving gifts began to be associated with the holiday.