Compact cameras using internal flash

There's a bit of a learning curve with underwater photography which means your first attempts will probably be disappointing.

The Problems

New underwater photographers often shoot like they do above water, using the camera on Auto and trying to capture moving schools of fish or scenery from too far away. The results will most likely be disappointing.
Compact cameras have slow autofocus and the inbuilt flash is very close to the lens.
All these factors reduce the odds of capturing decent images.

The Solutions

Move closer
This is the most important tip for improving your underwater results. Water is not as clear as air. The more water between camera and subject the worse the results will be. You need to be within touching distance to get sharp colourful photos with good contrast.

Pre Focus
Compact cameras are notorious for slow auto-focus and this gets much worse underwater. You can increase your odds of capturing a focused shot by pre-focusing. Half press and hold the shutter button to allow time for auto-focus to work, compose the picture, then press fully to take the photo. The shutter lag will be much shorter if the camera is already focused.

Under expose
Underwater shots look better if they are darker than above water. You can dial in some underexposure or "minus EV" if your camera allows. -0.7 to -1.0 will be about right.
Eventually you can learn to use Manual exposure settings for total control. Start with 1/125th and f/8 with ISO 200 and adjust from there.

Tips for best use of built-in flash

Wide angle scenic shots. (Subject more than 1m away)
The problem here is that your built-in flash is too close to the lens, it will only light up the suspended particles floating in the water in front of your subject. A small flash cannot penetrate more than about 1m through water so it's best to turn off or cover it. Use the widest zoom setting and get as close as possible to minimize the water between you and your subject. Aiming horizontally or up towards the surface usually gives much better results than down towards the seabed.
If you have a red underwater filter it can reduce the blue-green a bit. If not you will have to be content with blue green wide angle photos. You can adjust them on the computer later to some extent. Converting to black and white works very well with wrecks.

Close up (Subject 30 to 60cm away) Compact cameras can produce good colourful results in this focus range as long as the water is pretty clear. The flash will bring back colours but may also light up all the suspended particles in the water.

Macro (Subject 10 to 30cm away)
The potential for really sharp colourful shots is greatest in this range because haze and backscatter are minimized. However the close subject can introduce some extra problems.
The flash may be too bright when this close and the lens port may cast a shadow over the subject.
A diffuser or translucent plastic cover over the flash will spread and reduce the light output possibly improving results. The diffuser may need to sit out 1 or 2 cm in front of the flash to spread the light around the lens port and on to the subject.

To sum it all up
Move closer - I mean really close, like 50cm or closer.
Wide angle - Turn the flash off. Adjust exposure compensation to underexpose a bit.
Close up - Should work well
Macro - May need a flash diffuser.
Pre focus - Half press to focus then shoot.