Why do we light an eight-branched Menorah on Chanukah, when the Menorah in the Holy Temple had seven branches?
The number seven represents the natural order; the number eight represents the supernatural.
We light an eight-branched Menorah on Chanukah to commemorate the miracle of the small jar of oil, which was enough to kindle the Menorah for only one day, but burned for eight days.
However, the fact that the Temple Menorah had seven branches, while a Chanukah Menorah has eight, also symbolizes the different functions of the two candelabras. The Menorah in the Temple was kindled indoors, by day. Their purpose was not to provide illumination for the Temple itself, but to serve as a beacon, calling all the people around the world to come to the Temple and bask in the divine presence.
The Chanukah lights, by contrast, are kindled at night, and are placed near a window or door, to light up the outside. They are lit during a time of divine concealment, when we cannot simply go to the Temple to draw divine inspiration. Rather, we must go out into the darkness of night and reveal the hidden beauty and light that lies buried within the darkness itself.
For this reason, the Menorah in the Temple had seven branches while the Chanukah Menorah has eight. Seven represents the natural order; there are seven days of the week, corresponding to seven divine attributes with which G-d created the world. The number eight represents a power that is above nature.
The Temple required only a seven-branched Menorah. The power of seven was sufficient during a time when the divine presence was manifest. On Chanukah, though, we need the power of eight. We are charged with the task of illuminating the darkness outside, and transforming it, causing it to give off its own inherent light, hidden there since before creation. For this, the power of seven is not enough, and we need supernatural powers, represented by the number eight.