Three general types: rim, disc, and drum.

Drum brakes are most commonly seen as the well-known "coaster brake", lever actuated variants are very popular in many nations in their commuter or utility bikes.

Rim brakes are the most common type of brake in the US. Two composite pads that are clamped tightly to the rim through use of a lever acutated caliper. The caliper designs go from single-pivot calipers, dual-pivot calipers, u-brakes, v-brakes, roller cam brakes, cantilever brakes, and probably a few more variants I can't remember. Each type has their own advantages, and suck in their own unique and totally special way.

Disc brakes are the last type, instead of a rim, there is a metal disc attached to the hub of the wheel. This increases strain on the spokes, but allows for a far more powerful braking system overall. There are cable-actuated (mechanical) and hydraulic versions availible. One of the greatest advantages of disc brakes is it's stopping power when wet/muddy.