When animals are killed, their carcass can be skinned before being cut for meat. You do this by using the "Hideworking" skill menu (ALT+h) to skin an animal, then butcher for meat.
The quality of the acquired skin depends on how many and what kind of hits the animal has taken on its body. Piercing and slashing attacks destroy the skin the easiest so it might be wise to use blunt attacks to finish the animal off, if possible, or avoid targeting the body, instead aiming for the head for example. Damaged skins produce lower quality and lower value furs/leather.
The skin first acquired is fresh, but spoils in a few days, unless cured, tanned or stored in a cellar. Curing extends its lifetime. But tanning is much more useful, turning the skin into leather or fur, depending on the animal the skin comes from. Tanning use fats, which can come butchering a carcass, or barks, which come from peeling alder or rowans.
Animals that yield leather are cows, pigs, dogs, black grouse, boars, sows and snakes.
These animal's skins become furs when tanned: Sheep, Bear, Badger, Glutton, Beaver, Reindeer, (Artic) Fox, Elk, Wolf, Lynx, Polecat, Weasel, Pine-marten, Hare, Ermine, Squirrel. However, if leather is desired, the skins can be de-haired after cleaning and then the tanning process continued as usual to yield leather.