Protective Housing
Before you enter the pool or the ocean, do your research and buy underwater housing for your camera. You must balance the cost of your camera and the cost of the housing. If you are using an inexpensive point-and-shoot camera, then you may not want to spend a lot of money on the underwater housing. Test the underwater housing in your bathtub without your camera in it to make sure it is water tight.
Time of Day
Above water, photographers usually wait until early morning or late day to photograph outside. When photographing people underwater, you will want the sun directly above the water, shining as much light into the water as possible. Therefore, aim for around the noon hour.
Movement
Although it may be difficult with the water moving all arround you, remain as still as possible when photographing underwater. Since water absorbs a lot of light, set the ISO -- an indication of the sensitivity of the image sensor -- to a high setting before loading the camera into its protective water housing. Higher ISO pictures will contain more digital noise than lower ISO pictures. The higher ISO will allow you to use a faster shutter speed.
Panning
As people swim past you, move your camera to follow them as they go by. This technique is called panning, and when done correctly, it results in the subject being sharp and in focus and the background having a slight motion blur, which enhances the feeling of movement.