Most of my information came from MyJewishLearning.com and their Hanukkah section.
In 164 BCE, the Greeks were forcing assimilation on the Jewish people. The Greeks took over the Temple and started doing awful things, like sullying the altar by sacrificing pigs. A family by the name of Maccabee started a rebellion against the Greeks, and eventually won. Unfortunately, they were unable to clean and re-dedicate the Temple in time for the high holiday of Sukkot, so they decided to delay it a couple months. This is why Hanukkah lasts 8 days....because Sukkot does, as well. By this theory, Hanukkah is a celebration of the victory of a passionate, but weak, few against a mighty many.
One of the popular Hanukkah stories is that, when they were rededicating the Temple, they found only one jar of kosher oil, which was enough to light the traditional menorah for one day. It would take 8 days to make more, so they lit the menorah, and God miraculously allowed the oil to last until more was ready. By this theory, Hanukkah is a celebration of a supernatural miracle.
Hanukkah is about miracles and Jewish identity. Not everybody believes the legend of the oil, but the successful rebellion of the Maccabees is much easier to swallow. I think that both stories are important, and each ultimately boils down to restoring the ability to observe Jewish customs.
I think that people need to believe that the impossible can happen with supernatural intervention. If you can believe in that, then you can always have hope, because even when things are as bleak as they can be, God can change it. It doesn't mean that he will, and it doesn't mean that if he chooses not to intervene that you're not a righteous person, but it does mean that deus ex machina is possible.
The other side of the coin is the more people-based miracle; the miracle of a few people defeating a well-trained, well-staffed army. These are more along the lines of the miracles that happen every day, where people work as hard as they can, and with God's help, they are able to overcome obstacles. It's important to remember that God doesn't just hand you anything on a silver platter; you have to do your part.
Even though one of the themes of Hanukkah is Jewish identity and the preservation of a unique culture against the pressures of assimilation, I'm going to continue to celebrate it alongside Christmas.
Before you think I'm just too rebellious, let me explain.
While I very much believe in God in a way that I think is fairly Jewish, and I have found nothing in Judasim with which I disagree, the fact remains that my family will always celebrate Christmas. On December 25, my mom and dad will expect me to be at their house, sitting next to their Christmas tree, and going to a family dinner. That's our tradition, and it's a tradition filled with love and bonding. I'm not giving that up, even though the Christ part is much less significant to me. We were never into the mass part of it, anyway.
I will have Hanukkah in my own home, and I will appreciate it for what it is.