To snitch is to tattle on someone, and a snitch is someone who tattles. If you snitch on your brother when he "borrows" five dollars without asking, he might call you a snitch — but, on the bright side, he'll probably have to pay you back.
The oldest meaning of the informal snitch is "to betray" or, as a noun, "informer." This probably stems from 18th-century underworld slang, in which snitch meant "nose" — perhaps because a snitch is really nosy. A newer definition of snitch, "to steal," is thought to be a variation on snatch, and it first appeared around the turn of the 20th century. In the Harry Potter books, a snitch is a winged, golden ball used in the game of Quidditch.