Waves are formed by wind blowing on the surface of water. The stronger the wind, and the larger the area it blows on, the bigger the waves are that form. Water molecules closer to the surface move faster than water molecules at depth. This means that the top "layer" of water eventually overruns the lower "layer" of water, creating a wave.
The surf zone (where waves break) is determined by the direction of the swell and the contours of the bottom of the beach. The surf zone forms when the depth of the water is roughly 1.3 times the height of the wave. So a five foot wave will begin to break when the water is about six-and-a-half feet deep.
In order to catch a wave, a surfer has to accelerate up to the speed of the wave (wave velocity). The surfer must place himself further than the position where the waves break, and ensure that his distance from any approaching wave allows him to paddle sufficiently so that he can match his velocity to that of the wave.
After accelerating up to the wave speed, the rider pops up to a standing position and begins planing over the surface of the wave. Since the wave becomes steeper as it advances, the surfer has to move towards the front of the wave where the acceleration is less. Advanced surfers don't simply ride waves straight out to the beach, but travel across waves, in a path that is parallel to the crest. The surfer rides along the position of the wave where the water is starting to rise just before breaking. A surfer can steer a surfboard by shifting their weight between the balls and heels of their feet, along with the help of the fins attached to the bottom of the board.
The native inhabaitants of Hawaii surfed straight out to shore on huge, wooden surf boards that had no fins. When surfing became popular in the USA, modern boards that were a lot lighter, with fins, were developed. In the 80's the short board became popular, and surfers started doing skateboard-inspired jumps on the water.