A corpse is what is left of a creature once dead. The corpses of vermin are called remains.
A corpse has several levels of decay, it will begin as "(creature or name) corpse", progress to "rotten (creature or name) corpse", "(creature or name) partially decayed corpse", and then to "(creature or name) skeleton", and will remain a skeleton indefinitely. Alternatively, if the creature had had parts ripped or cut off before death, the corpse will be described as "mutilated corpse" and "partial skeleton" in respective descriptions, and the parts (if still on the map) will decay into a type of bone depending on the part. Note: this is the only way to get useable bones from creatures dwarves refuse to butcher according to their ethics (unless you modded the ethics). Cutting a goblin up using serrated blade traps or throwing them down extreme heights (such they explode) will generate useable goblin bones and skulls. Already-dead corpses can be reanimated with the help of a necromancer, or naturally if you are in any type of evil embark, and trapped. Remains of vermin, however, simply progress from "(vermin) remains" to "rotten (vermin) remains" before simply vanishing.
Butchering a corpse produces quantities of meat, prepared organs, bones, a skull, skin, and nails/hoofs, as well as cartilage, nervous tissue, and other types of useless byproducts. Body parts can be butchered, presumably for whatever tissues/organs were in the part. "Butchering" a skeleton produces only a skull, bones, cartilage and other non-decaying tissues. The corpses of very small animals, such as of ravens, currently cannot be butchered.
Corpses will produce miasma in subterranean areas, but not above ground.
As of v0.34.xx, corpses can be raised from the dead. This will occur in some evil biomes if the creature is killed inside an evil biome (if killed outside of the evil biome on an embark with part non-evil biome and part evil biome, the corpse will not rise) or if it is raised by a necromancer. Corpses may also be risen from the dead by mummies, though mummies will only be encountered in tombs in adventure mode. Severed body parts will be able to rise if they have either a grasp tag (hands and possibly other body parts) or if they still have a head attached, so it may be advisable to kill risen corpses with blunt weaponry to avoid swarms of body parts. Corpses and severed body parts will not rise again if they have been butchered, their skin tanned and their hair spun into thread, or if they have been exposed to magma. However, due to what is presumably a bug, magma and fire damage currently will not destroy an animated corpse.