Skinning

Skinning allows players to gather leather and hides by skinning animals. Leather and hides are used for leatherworking. Leather is also used for some blacksmithing, engineering, and tailoring recipes.

Becoming a Skinner

First, find an NPC who can teach you the skinning profession (look for the word "<Skinner>" under the NPC's name). They will likely be somewhere near leatherworking trainers. Ask a guard for directions. Now, you need to purchase a skinning knife. You can find skinning knives at merchants who sell Profession-related items. Place the skinning knife somewhere in your bag. You do not need to equip it.

How to Skin Animals

Only certain monsters can be skinned. When you kill a monster, select its corpse to see if it is skinnable. If the animal is skinnable, it will display "Skinnable," with a color to indicate the difficulty. Open up your abilities book ("p") and find the "Skinning" icon. Drag the icon over to your action bar. Right-click on the icon and target a corpse that is marked as skinnable. Your character will then begin to skin the monster. You can also just right-click on the corpse to skin it after it has been looted.

General Information

As your skill goes up, you become better at gathering higher-quality materials.
The text displaying "Skinnable" is colored the same way as other Profession abilities, indicating how difficult the task is (or whether it's even possible at your current skill level) as well as how likely you are to earn skill points.
You cannot skin the monster if your skill is not high enough. You will have to practice on lower-level monsters that are easier until you have enough skill to skin a higher-level monster.
If you kill the animal (or it's your turn to loot it in a group), you must loot it before you can skin it.
You can skin animals that have been killed by other players that are just lying around if they have looted the corpse. Generally it's polite to not skin the slain enemies of other player unless they are not also skinning. First watch to see if they skin a monster after killing it. If they walk away from the corpse, you can move in and skin the monster.
You cannot skin a monster killed by another player unless they loot all of the contents.
You can skin some critters that don't fight back such as bunnies, sheep, and cows.
To skin higher level hides and leather you will need to kill higher level monsters.
Beasts do not drop leather, you must instead skin them.
You can hold down shift and right-click on a dead monster to skin and auto loot them in one motion.
You can create higher level leather by combining lower level pieces of leather using Leatherworking recipes.
Leather
Here is a list of Leather that can be skinned listed from the lowest level to highest:

Ruined Leather Scraps
Light Leather
Medium Leather
Heavy Leather
Thick Leather
Rugged Leather
Core Leather
Hides
You can also skin hides which can be used in leatherworking. You can skin Light Hides, Medium Hides, Heavy Hides, Thick Hides, Rugged Hides, and various other hides such as Raptor Hides, and Shadowcat Hides. If you find something strange while skinning, look around for its use. You can also check the auction house to see if people are selling them.
Hides will be found more rarely when skinning vs. regular leather.

Alterac Valley
You can skin certain animals in the Alterac Valley battleground to help your side win the battle.

When you start out as a skinner, this formula doesn't apply exactly, but once you skill past 50 or so, to figure out the maximum level mob you can skin, divide yor skinning skill by 5.

Apprentice-level skinning allows you to skin up to a level 15 creature.

Curiously, beasts in the immediate starting area (Coldridge/Northshire Valley, Shadowglen, Ammen Vale, Valley of Trials, Red Cloud Mesa, Deathknell, Sunstrider Isle) can't be skinned.