“What?” I hear you say, “Isn’t the Menorah only lit on Chanukah?” No, our Chanukah “menorah” (AKA a “Chanukiyah”) is lit to commemorate a miracle involving THE Menorah, a five-foot tall candelabra that stood in the Tabernacle and, later, the Temple. The Menorah was accessed by the kohanim through a series of steps that stood in front of it. (The Kohein Gadol was not allowed to raise his arms above the tzitz, a metal plate worn on his forehead, bearing the name of God. Accordingly, he had to stand over the Menorah in order to arrange it.)
The reason for this mitzvah is that the glory of the Temple is enhanced by the light, causing it to be held in greater reverence by the people who go there. Like offering the sacrifices, lighting the Menorah was an activity performed daily that otherwise would have been prohibited on Shabbos.
Part of the service of the Menorah was cleaning the cups that held the oil and removing the ashes . There were seven branches on the Menorah; the six outer wicks faced the center. The six outer wicks could be lit from one another; if the center light went out, it was re-lit using fire from the altar. The Rambam sees the maintenance of the Menorah as part of the mitzvah, though some other authorities consider maintaining and lighting the Menorah to two separate mitzvos (perhaps based on Numbers 8:2).
The obligation to light and maintain the Menorah applies to male kohanim at a time when there is Temple service. It is discussed by the Talmud in Menachos, primarily on pages 88a-89a, and is codified in the Mishneh Torah in the third chapter of Hilchos Tamidin. It is #25 of the 248 positive mitzvos in the Rambam’s Sefer HaMitzvos.