Besides as an adverb means "in addition" or "moreover": It's Albert's birthday, and besides, you promised.
Besides is also a preposition meaning "other than" or "except": Who besides me is hungry?
Compare that with The person beside me is hungry. Beside is a preposition that means "next to," "near," "alongside."
A lot of people say something is "besides the point." They mean beside the point. When a statement is beside the point, it misses the mark and settles nothing.