Paddling a surf board correctly is one of the most fundamental aspects of learning to surf.  Your surf instructor can push you into waves for days and even tow you back out,  giving you an excellent surfing experience, leaning to paddle the board yourself is key to becoming a real surfer.  There are two paddling strokes, the butterfly and the crawl, both are swimming terms used to help a surfing student visualize how to paddle a surf board.  The trick is to get the surf board gliding on the water by not creating drag at the end of the stroke or paddle.  This is achieved by taking the arm fully out of the water as you move it as far towards the front of the surf board for your next stroke.