If your fish is unlucky and gets hooked in the gut or the gills, or if a hook seems stuck in the lip, don't try to remove it. Removing it may tear the fish's flesh and organs and exacerbate its injuries. Instead, simply cut off as much of the hook as possible and release the fish. After a while, hooks that are left in will rust and fall out or dissolve. For that reason, you shouldn't use hooks made of stainless steel or some other corrosion-resistant material. You also should avoid treble hooks because the extra two points increase the chances of hitting a blood vessel. They also take longer to remove.
That red snapper you're enjoying for dinner might well be tilapia. Learn more about fish fraud at HowStuffWorks.