This is a Red Bellied Watersnake. Snakes have a single lung which is very long. Taking a deep breath to fill it with air, they become quite buoyant. To dive, they fill the lung with just enough air to give them almost neutral buoyancy.

To move forward, they undulate their bodies horizontally, pushing against the water in basically the same way that they push against loose sand. The water is pushed backwards, and the snake is pushed forwards. This is the same way that eels, leeches, and many marine worms swim, and it is very effective.