Around 168 BCE, the Seleucid King, Antiochus IV Epiphanes, tried to Hellenize (force Greek culture on) the Jews, banning Jewish practices and desecrating the Temple in Jerusalem.  The final straw was slaughtering a pig in the temple to sacrifice it to their false god, Zeus.  A small band of Jewish rebels, led by Judas Maccabeus (the Maccabees), fought back against the larger Greek army.  After three years, the Maccabees won, reclaimed Jerusalem, and began to cleanse and rededicate the desecrated Temple in 164 BCE.  Tradition says that when the Maccabees went to relight the Temple's Menorah, they found only one small jar of consecrated oil, enough for just one day.  Unfortunately, it takes eight days to make new oil, but miraculously the oil burned for eight days and nights until new oil could be prepared.
It is also said that since they couldn’t celebrate Sukkot, this was like a second Sukkot allowing them to celebrate. Notice the 8 days? The second book of Maccabees also records the people celebrating with palm branches, ivy-wreathed wands, and leafy boughs - just like the four species used to celebrate during Sukkot. More than likely, everyone was fulfilling the commandment to celebrate Sukkot. 

Although it's not a commanded feast, Hanukkah is mentioned in the New Testament:
At that time the Feast of Dedication took place at Jerusalem. It was winter, and Jesus was walking in the temple, in the colonnade of Solomon.  (John 10:22-23)

This was a big event.  Jesus wouldn't have been randomly walking around the temple at that time.  Space was limited... we can  infer that He went there on purpose for this holiday.

You can also find the events recounted in the Books of Maccabees.

Hanukkiah/Menorah
We light a menorah on all eight nights of Hanukkah, remembering the miracle of the oil and rededication of the temple.  This menorah has nine spaces instead of seven, and is also known as a hanukkiah.

Oily Foods
We eat a lot of oily foods throughout the week to specifically remember the miracle of the oil lasting all eight nights.  Anything fried will work, like donuts, fried chicken, and latkes!

Playing Dreidel
During the rule of Antiochus IV, Jews were forbidden to study Torah.  They did anyway, but when officials approached, they would act like they were playing this game instead. 

Gelt
Gelt represents the plunder that the Maccabees redistributed to the people.  The Jewish people also minted their own coins after the victory, and this represents the political autonomy.

The most common traditions include lighting the menorah, eating oily foods, playing dreidel, decorating with lights, and giving gifts.  

Around here, we also like to do special winter stuff.  We enjoy hot chocolate while watching Hanukkah shows, eating homemade chocolate pudding while it's still warm, playing in fake snow, making edible snowmen from meringues,  pancakes shaped like menorahs, and getting donuts in our jammies!