Seatpost clamps are used to tighten your frame around the seat post. There are 3 main sizes, 28.6mm which is typically used on steel frames and finally 31.8mm and 34.9mm can be found on alloy and carbon frames.
There are three types of seatpost clamps. Quick Release, which allows the seatpost to be moved easily. Allen bolt which is more secure and generally lighter and also another Allen bolt version, which also has mounts to fit racks.