Use a Light (or Strobe)
Any photographer (land or underwater) will tell you that the key to a good photo, is good lighting. You can never have enough! This is even more important for underwater photography, as the water absorbs the light so much more than it does on land.
Many photographers opt for a combination of strobes and lights, as strobes are significantly brighter and more powerful than your standard light, and they do a much better job at freezing the motion. You can get away with very slow shutter speeds, while not having to worry too much about blur or camera shake.
Strobes can be a significant investment, so most beginners start with a good photo/video light. You want something with a fairly wide beam (to avoid hot-spots in your photos), and the brighter the better! The advantage of shooting with video lights is that you know exactly what the lighting is going to look like before you take your picture. Not to mention they are a lot smaller and less expensive than strobes.
Having a light will also help your camera focus a lot easier!
Get Close, and Then Get Closer
The closer you are to your subject, the more detail you will be able to capture. Your lighting will be more effective close up, and you’ll also reduce the risk of backscatter (the pesky particles that reflect the light between you and your subject)
Master Your Buoyancy
One of the biggest and most important skills to master for underwater photography is your buoyancy. Your photos will be sharper, and your dive buddies will appreciate you not stirring up the bottom!
Often when you are shooting downwards, your background will be comprised of sand, dirt, and rocks, which can clutter the image and does not allow your subject to stand out. Whenever possible, try to shoot upwards. This will allow you to isolate the subject, and really get it to pop!
Practice Good Photographer Etiquette
Be considerate of other divers (both with and without cameras). Wait your turn to take pictures, and try not to crowd other photographers. There is plenty of life to go around! On the flip side, if you find a rare critter, keep in mind other divers might like a chance to take a look. Take a few shots, and then allow someone else a turn. Be extra careful when moving away not to stir up any silt for the other photographers/divers. Once you’ve moved away, take a moment to review your photos and make any necessary setting changes before going back for more pictures.