There is a review of underwater compact cameras and the systems at:
Underwater Photography Guide
I have the Canon G11 with an Ikelite Case and Strobe, which is Canon's most compable compact camera. It does a pretty good job. On land, I use a DSLR.
I would strongly suggest getting a camera with RAW capability. You will need larger memory cards. But RAW gives you much more flexibility in post processing. It also allows you to modify the white balance to your heart's content after the fact. In JPEG, you are stuck with what you shot. On land that usually is not a big deal. But white balance is tricky underwater.
If you get a camera with RAW capability, you are going to get into the most advanced point and shoots. You will be hitting 10 megapixels anyway.
If you are diving to any depth at all, you will probably want a strobe. Otherwise your photos will be nothing but shades of blue and black. You need a strobe to add back the colors. Now the camera does have its own flash, but it is not too helpful. The reason for this is backscatter. The camera's strobe will light up every little particle in the water making your shot look like it was taken in a blizzard. You need a strobe that is shot off to the side so the reflected light does not bounce back into your camera image.
There are some pretty modestly priced systems. The review I cited will probably give you an idea.
However, if you are going to take your photos in clear water and less than 20', you should do fine without a strobe. It also helps if your subjects have a bit of white on their suits. My diving partner has some white lettering on her wet suit. I just tell the post processing photo that the white is white and it corrects the white balance in a single click.
I would also suggest a pretty decent case. I had an earlier camera with a pretty inexpensive case. It flooded once. That ruins the camera. It was pretty hard if not impossible to tell if the case was really sealed. My Ikelite case (which is considered one of the most modestly priced) is far more secure.
