The celebration includes lighting candles on a hanukkiyah (a type of menorah). On the first night, one candle is lit, the second night two candles are lit, the third night three candles are lit, and so on until the last night when all eight candles are lit. Also during the eight-day timeframe, people eat food that has been fried in oil. Two of the more popular foods for this occasion are potato pancakes (called "latkes") and jelly doughnuts (called "sufganiyot"). Lastly there's plenty of time for games; one such game is played by spinning a dreidel (a four-sided top marked with a different Hebrew letter on each side) and gambling on which side it will land.
Hanukkah took on greater emphasis in the Western world in the 20th century. According to George Robinson, "When Jews found themselves coexisting in relative peace with their Christian neighbors in post-World War II America and Europe, this mid-winter holiday became a sort of Jewish counterpart to Christmas...".
I saw this when I growing up: kids had a sort of tongue-in-cheek envy of Jewish children for having eight days of gifts while the Christian children had only one. This clip of Adam Sandler singing his infamous "Hanukkah Song" highlights the point:
What's the Celebration About?
The Jewish people had returned from their exile in Babylon and, around 516 BC, completed a new temple in Jerusalem, called the "Second Temple" (around 19 BC it came to be known as "Herod's Temple," after Herod the Great renovated it as part of Rome's foreign/domestic policy). The temple had a central significance for Jews because it was where God, Yahweh, had promised to dwell and meet with them.
Meanwhile, the Seleucid Empire of Syria was gaining strength and taking land. Antiochus IV Epiphanes ruled the empire from 175 BC until his death in 163 BC. Interestingly, he claimed the title "Epiphanes" which was shorthand for "God Manifest." Antiochus sought to use Judea as a buffer against Egypt. Desiring the people's sole allegiance, he took over the Jerusalem Temple on December 25, 167 BC. The Torah scrolls were burned, worship of Greek gods (including himself) was required, and unclean animals were sacrificed on the altar in the temple. This is thought by many biblical scholars to be the Abomination of Desolation mentioned in the Old Testament book of Daniel.
Whether or not that particular reference is accurate, the book of 1 Maccabees sums up this tyranny of Antiochus and his troops:
"They erected a desolating sacrilege on the altar of burnt offering. They also built altars in the surrounding towns of Judah, and offered incense at the doors of the house and in the streets. The books of the law that they found they tore to pieces and burned with fire. Anyone found possessing the book of the covenant, or anyone who adhered to the law, was condemned to death by decree of the king" (1 Macc 1:54-57 NRSV).
Rather than rallying around Antiochus' cause, some of the Jews refused to go against their ancestral laws. Instead of becoming supporters for Syria, some of the resistant Jews became martyrs for Yahweh. One such rebel group was comprised of Mattathias Maccabaeus and his five sons, collectively called the "Maccabees."
Hanukkah, which literally means "dedication," then, is a time to remember and celebrate that the temple - the place where God has promised to dwell with his people - has been eradicated of its abominations and consecrated again to Israel's God.
The alternate title of "The Feast of Lights" comes from a story that circulated at the time. As the story goes, the Maccabees demolished the polluted altar of the temple, as well as the other tainted objects. They built a new altar and replaced the objects. In the process they found a small amount of holy oil for use in the sacred lamps of the temple, but it wasn't nearly enough to use for the rededication ceremony that was to last eight days. The priests, filled with faith, decided to light the oil for the first day. Miraculously, the next day there was enough to continue on. The oil kept burning through the next day too. So it continued until the full eight days had been observed. It's in memory of this miracle that Hanukkah is celebrated each year by lighting candles on each of the eight days .
David DeSilva summarizes the twin functions of "dedication" and "lights." He says, "This festival kept in remembrance the dangers of foreign domination, the heroism of the martyrs who died rather sacrifice fidelity to the covenant with God, and the remarkable successes of the Jewish revolutionaries who routed superior Gentile forces. It also spoke eloquently of God's continuing fidelity to God's holy place".
And that is what Hanukkah is all about.
 George Robinson, Essential Judaism: A Complete Guide to Beliefs, Customs, and Rituals (New York, NY: Pocket Books, 2000), 111-112.
 The first temple was called "Solomon's Temple."
 N.T. Wright, The New Testament and the People of God (Minneapolis, MN: Fortress Press, 1992) 158-159.
 Robinson, 112.
 David A. DeSilva, An Introduction to the New Testament (Downer's Grove, IL: InterVarsity Press, 2004), 79.