Fish float because of a specialized organ called the swim bladder. It's essentially a bag full of gas that balances out the denseness of the fish's other bodily tissues so that it can remain buoyant as it swims.
1 Additional Answer
The reason that fish don't just sink down to the bottom of the ocean is because they have and swim bladder. This is an expandable sac inside the fish that they fill with oxygen to make them selves rise.