So what is Hanukkah anyway? It is the commemoration of an even that occurred after the time of the Torah.
During the 2nd century, Israel was divided up by the Greek empire. The emperor prohibited all local religions and tried to make the locals worship the Greek gods (and the emperor). The Jews insisted that their G-d was the only G-d and resisted. These rebels were called the Maccabees. They fought back and won their religious freedom. But the Temple of Jerusalem had been turned into a Greek shrine. It had to be cleansed and rededicated for Jewish worship.
To be cleansed the menorah had to be lit for 8 days. It was found that there was only enough oil for 1 day. They lit the menorah and the menorah remained lit for 8 days.
Now Jews commemorate this miracle by lighting candles in a hanukkiyah. A hanukkiyah is a special menorah with places for 9 candles. The 9th candle is the one we use to light all the others (it is called the shamash which means servant or sun). On the 1st night of Hanukkah, we use the shamash to light 1 candle. On the 2nd night we use the shamash to light 2 candles. We add another candle each night until on the 8th night all 9 candles are lit.
Many Jewish holidays also have certain foods associated with them. For Hanukkah, latkes (potato pancakes) and doughnuts are popular because they are both fried (fried in oil, oil for the menorah... see the reasoning).
Gift giving is a more recent development for the holiday. Yes, our family exchanges gifts but we also encourage the children to give tzedakah (giving money to charity). The gift exchange has made the transition easier for the children from Christmas (celebrating from a secular aspect).
Now for your viewing pleasure... a little fun... one of my favorite Hanukkah videos!