All corpses will rot away, eventually. All corpses will leave behind their posessions where they died. Corpses left to rot in an enclosed space, such as your fortress, will emit miasma. After a time, the flesh will rot away, leaving behind bones and a skull.
The corpse of a dead dwarf must be buried. To avoid unhappy thoughts, the sooner it is placed in a burial receptacle, the better. Any dwarf that dies will make unhappy thoughts for anyone it is related to, but left to rot where it is, all dwarves passing by will have unhappy thoughts.
The corpse of an animal can be valuable to your dwarves. When brought to the butchery, a butcher will dice up the corpse into meat, bones and fat. Depending on the animal, raw hide and skulls may also be rendered.
The corpses of enemies will often contain their clothing, armor and weapons, if any. Your dwarves are automatically set to retrieve these items and put them into your containers. This can be a very bad thing in the middle of a siege, as dwarves not locked away or do not have hauling jobs turned off will walk into the middle of a battlefield to retrieve these items.
If the corpse was involved in a fight before death, it is possible for parts of the corpse to be scattered around. Hammerlords and the like are notorious for blasting apart bodies in one swing. Should a corpse be severed from its parts, all the equipment you'd expect to be on the limbs will usually be located where the limb fell.