We went to nearby big town yesterday and bought a whole pomfret and a whole Spanish mackerel (cleaned!). I watched several videos about how to fillet the pomfret, assuming it was a "round" fish, rather than "flat." It looked easy enough on the video. Not! When I put my sharp knife along the ribs, which looked like the top of the pomfret and attempted to enter the fish, it did not give at all! Eventually I produced two fillets, but rather clumsy, and I suspect I wasted a good bit of the meat in the doing. What are the tricks?
Gio's reference will get you there with just about any fish - they're all built the same. Here are some extras.
Instead of slitting out the pin bones (if necessary) as described, drape the fillet over a large, rounded bowl. This will pop up the pin bones and you can remove them with needle nose pliers - run your fingers over and you'll easily feel any that didn't come come visible.
The demo shows the rib cage being removed with a single cut. I like to do just 3 or 4 ribs at a time using the next rib as a guide and this saves a lot of meat.
If you need to skin a fillet, the easiest way I've found is to use a cheap, serrated bread knife. Lay the piece flat on your cutting board, start at the tail and work forward with the blade at an ever-so-gentle a downward angle. As you advance with your cutting hand, advance the fingers of your your free hand just behind the blade, keeping the skin flat. The blade is flat, its sharp edge is raised a bit, and it should just clears the layer of skin.
Remove the gills (if present) from the head and discard them; chop the frame into usable lengths; add everything left to a pot of water, and simmer until the flesh just turns opaque. Strip the flesh off the frame and return the bones to your pot and simmer for about 10 minutes or more for a fish stock. You'll be surprised at what you can recover for a fine chowder or(?).
As EricMM said, the filetting blade should be razor sharp - I'm amazed at how resilient fish hide is.
When I fillet a fish, I make the first slash behind the head. Do it at an angle, so the cut begins just behind the top of the skull and ends behind base the pectoral fin. (The fin that corresponds to your arm.) Try not to cut too deep...the blade should stop where you feel the ribs. You may also want to make a vertical cut across the base of the tail, but that isn't necessary. If the fish is small, say 5 lbs and under, I simply angle the blade behind the head cut and slide it, with a sawing motion, all the way to the tail, staying on top of the rib cage. This produces a very neat fillet. If the fish is larger than 5 lbs, I make that same initial cut behind the head, but then I make a long cut along the dorsal fin, just through the skin. I then slide the knife along that same cut, working downwards to the ribs. Once past the ribs, I cut down to the spine and finish it off the same way as on a small fish. It goes without saying, though, that to get a neat fillet you need a very, very, very sharp blade! One that is also very thin and flexible (unless your filleting a 10 lb+ monster, where a stiffer blade may help).