The first thing to do is make a cut from near the gill down to the backbone. There's a lot of bone and hard cartilage directly over the gills, so you'll have to find the first soft (fleshy) point behind the gill. Holding the fish's head, slice down into this soft spot at a slight angle towards the tail. Look at the anatomy picture - all fish are built roughly the same way.
Slide the knife through this cut until it passes from the head out the other side of the fish, near the pec fins. You need to turn the knife blade so that it's lying flat to the backbone.
Use a slight - very slight - sawing motion to slice the meat from the backbone towards the tail. Make a slight slice, and adjust the knife to make sure it's actually scraping the backbone. Keep adjusting and slicing until you've reached the tail. Don't separate the meat from the tail, though. Leave just enough skin intact to hold it in place. Flip the fish over, keeping the nearly separated filet underneath the body. This provides support for the backbone while you're cutting the second filet. If you remove it too early, the backbone sags to the table and makes the second filet more difficult to remove cleanly. When you've made both cuts, remove the filets from the tail (and hence the body).
If you're going to throw the fish carcass into the bay (which is just fine - lot's of things are just dying to munch on those great bones), make sure you poke the eyes to remove the fluid. Otherwise the fish will float to the surface.
Also, please break or cut up the carcass into several pieces. If a pelican gets one of these intact bony structures stuck in its pouch, it could very well be seriously injured or kill the hapless bird.
At this point, you'll have two, (hopefully) whole and well-manicured filets - with skin on one side.
In most cases, you'll want to remove the skin. One exception is redfish - try leaving the scales and skin on the fish, and roasting them on the barbeque skin side down. For some reason, the armor-like skin and scales hold moisture in the filet and makes for an incredibly tasty, juicy dinner. Simply sprinkle a little garlic on the meat side before you start, and add a little olive oil or butter (better) when it starts to get hot. Test for doneness - when it's ready it will flake off the skin easily.
To skin the filets, grab the tail side with your fingernails, and slide the knife between the skin and the meat for about four inches or so. Cut a small slit in this skin (along the long axis, tail to head) and stick your finger in it. It will serve well as a handle for holding the skin while you slice the meat off.
One last thing. Don't try this with a dull knife. Sharpen the knife before you start, and keep a steel handy for touching up the blade during the process. The knife will be dull after only two fish, so keep it sharp.