Here's how to tell the difference:
Preposition: When "besides" is followed by a noun or pronoun, it acts as a preposition.
* Example: *Besides the cake, we also had ice cream.* ("Besides" introduces the noun phrase "the cake").
Adverb: When "besides" is used alone, it acts as an adverb, meaning "in addition to that" or "moreover."
* Example: *I don't like this movie. Besides, it's already late.* ("Besides" introduces an additional point).