Skinning is a feature which helps players to obtain leather and other Creature Products, which can then be used to receive Outfits and Addons. To use this feature you need an Obsidian Knife, which can be obtained in the Obsidian Knife Quest.
Most of the time, you won't get Creature Products if you skin a body. If you failed, you will see a yellow spark. If you see green sparkles, the product of the specific creature will have appeared in the same container as your Obsidian Knife. The chance of getting a dead body skinned successfully varies from creature to creature.
Since the 2015 Winter Update corpses can be skinned immediately after the creature died, so you do not need to wait 10 seconds anymore before being able to skin a corpse. As corpses can only be skinned in their first state of decay, this gives you a time frame of 300 seconds during which a corpse is skinnable. After skinning a body, it will automatically rot one level (to the second state of decay) to prevent multiple skins on the same corpse.
There are two exceptions to the two rules mentioned above: both the corpse of the Lord of the Elements and of The Mutated Pumpkin can be skinned multiple times, at a 100% success rate (but still only once per player).