There are two main types of snowboard bindings: conventional and step-in.
Conventional, or strap-in, bindings are the most common type and are preferred by most advanced riders. Strap-ins, as the name suggests, lock the rider's feet into place with straps the tighten down over the boots. Typically, there are two straps, a heelstrap and a toestrap, however, some other variations do exist. Strap-in bindings usually have a high-back made of plastic or other material which rests against the rider's ankle & calf for enhanced leverage and responsiveness.
Step in binging systems allow for added convenience, quickly locking onto some sort of metal connector on the bottom of the rider's boot when he/she steps into the binding and releasing when a lever is lifted. Step-ins may or may not have high-backs, but most do not. They are quite popular amongst beginning snowboarders for their convenience. Because they are generally less responsive and tend not to hold the foot in place as securely, step-ins are rather unpopular with experienced riders.