Very good, Indie.
I think they are generally skinned, the fur (or feathers) preserved, then the skin stretched around a suitable shape, I think traditionally made of wire, straw and clay.
I think the head may use the animal's own skull. There are catalogues with hundreds of slightly different glass eyes to make an exact match (
http://www.watdon.com/watdon_home.html, and click on Taxidermy).
Fish are done by casting them in plaster, and I think just painting them, presumably with some bits of the jaws attached.
It's all very skilled -- you can see how difficult it is by how unlifelike most stuffed animals look.