This is the only single holiday that is celebrated for eight days. While both Pesach and Sukkoth are celebrated for 8 days outside of Israel, they are, in essence, 7 day holidays with one day added on for those in exile. Chanukah is eight days all over the world with no additional day added. Eight is a highly significant number in Jewish thought. It represents a level of spirituality that rises above nature. That is why a brith [circumcision] is performed on the eight day.
The small jug of pure oil that should have sufficed only for one day burned for eight days to allow enought time for more pure oil to be made. That always leads to the question of why we celebrate the miracle for eight days, when the miracle was really only for seven. There are various answers for that. The most common are that finding the oil at all was a miracle or that the additional day is for the miracle of our victory in battle when our forces were so outnumbered. But the number eight is also what this holiday is all about -- reaching beyond the natural order in our spiritual aspirations.