I suppose taking photos in the bathtub could be considered underwater photography, but it usually means photos or films taken in larger bodies of water. The amount of shipwrecks, animal and plant life under water is very hard to comprehend, so it's understandable why photographers want to explore the seas and oceans and preserve their findings on film.
Most underwater photography is done by scuba divers, but snorkelers and swimmers can also enjoy it along with high tech devices such as mini-submarines. Perhaps the most famous instance of underwater photography was the finding of the Titanic.
If you want to shoot wildlife underwater it can be difficult since it and you are usually moving simultaneously. Regular cameras can be used underwater, but they need to have special watertight housings to cover them. The deeper you go down, the darker it gets, so you will need a flashlight or your flash should be mounted on an arm.
If you are just trying this technique for the first time you may want to use a disposable underwater camera until you get the hang of it. Specialized equipment is available for underwater photography and it should be comparable in price to regular equipment.