The 2 types of mounts are a "skin" mount and a replica mount. The skin mount, as the name implies, uses the skin of the fish and is "stuffed" so to speak and then recolored to bring back the living colors. Just be sure to take a picture of the fish fresh out of the water and record all the measurements: length, weight, girth, etc. A replica is a "blank" the size and shape of the fish, made of Styrofoam, resin or fiberglass. The taxidermist then uses your photos and measurements to create an exact "replica" of your fish.
Either way you go, make sure you check out samples of the taxidermist's work -- with fish mounts, the quality of the painting and the paints and sealers used matters much more than whether or not there's a fish skin in the mount. If you have photos of the fish fresh out of the water they can capture the colors of the fish more accurately.