Strictly speaking, the "rings" of Saturn are not really rings, as they are not solid and are nothing more than bits of dust and ice, held in place by gravity. However they certainly have the appearance of rings.
Deciding how many rings there are is a matter of opinion, but here is the generally accepted breakdown.
There are two "main" visible rings, known as the A and B rings. (The A ring is outermost, and the B ring is closer to Saturn and wider). These are separated by a gap known as the "Cassini Division" which is thought to be caused by the moon Mimas.
There is also another large ring called the C ring which is further inwards than the B ring. Although this ring is wider than the A ring, it is also much darker so is less noticable.
Other rings are a matter of opinion, but it is generally accepted that there is an F ring (outside the A ring) and a D ring (inside the C ring). There are also rings further out known as the "dusty" rings, these can be subdivided many ways, but calling them the E and G rings is fairly standard.