Rubber band-powered wooden boats have a flotation surface (usually flat), that is cut in whatever shape the builder wants; making them drop-shaped with the narrow end forward improves speed and reduces drag. The back of the flotation surface has a notch in it where the paddle wheel goes. The propulsion mechanism is two wooden slats with a notch cut in them. Putting them together so the notches slide together makes a paddle wheel, which is then threaded through the rubber band and stretched over the two flanges at the back of the boat.
The rubber band acts like a coiled spring and stores energy. When the wheel is let go, the rubber band relaxes and spins it. When put into the water, the paddle wheel will push the boat forward (or, if you rotate it the other way, it can pull the boat through the water in reverse).