The first and most important element of success with underwater photography or underwater digital photography is your choice of camera. Obviously, you need to purchase a camera designed specifically for underwater photography or purchase an absolutely watertight housing for your camera.
Personally, I recommend starting out with a disposable or inexpensive underwater camera to see if underwater photography is for you.
Before you go scuba diving, it's a good idea to practice taking photos underwater in a swimming or wading pool. You can drop some heavy objects that will sink down to the bottom of the pool and then float some lighter objects in the water, as well. Try shooting from different angles.
For beginners who don't have professional lighting equipment, it's a good idea to shoot photos within the first few/top meters of water. If you want to shoot photographs of objects further down, select an area of shallow water where the light will be better.
Initially, it's also easiest to shoot in clear water, as well. Murky water or water with lots of sediment makes it difficult to see your subject. It's also easy to stir up particles of the sediment that can show up in the photograph as they float through the water.
Use an underwater photographer's mask so you can see properly while diving and clearly see the image you're composing. Then be sure you don't just snap a photo. Instead compose a picture. Just like other types of photography, underwater photography should be deliberate and focused.
Follow the rule of thirds, diagonals, or other commonly accepted guidelines for composition when creating underwater photographs, just as you would for other photos.
Look for colorful, interesting subjects to photograph. Then get as close to your subject as you can and use the flash. Remember, the deeper underwater you go, the less light you have, and thus the less colors will show up. A flash revitalizes your color.
Try to change the angle of your camera as necessary so your background is a solid color or a neutral color so that your subject really stands out.
Try close up or macro photos with a macro lens. Then experiment with a telephoto lens to get close-up shots of your subjects without spooking the fish or other underwater creatures. Try different lenses to get different effects.
Prepare your camera properly before immersing it. This means making sure you have a fully charged battery, your memory stick is properly installed, the lens is clean, the lens cap is off, your settings have been selected the way you want them (be sure to set your auto focus point before you take the camera underwater), and the camera housing does not leak. Also test any flash or strobes to ensure everything is working properly.
Many of the principles of "normal" photography apply to underwater photography, as well, but it is particularly important to plan ahead for underwater photography, as it's much more difficult to change a memory card or a battery or do some of the other things that can be quickly adjusted outside of the water. Take your time to familiarize yourself with your camera, plan for your photo session, and then have fun experimenting under water!