Skin

A Skin refers to the texture that is placed onto a player model or mob.

The skin is divided into areas that act as the surface area of the character (For example, there's the front head area, left leg area, etc.). A skin will only allow solid color; transparency is not allowed on the skin file except on the second head area, which is transparent by default. The second head layer can be used to give the character glasses, hats, or other accessories (even a bigger head).

World Textures
A skin can also refer to other textures in the game, such as block textures, item sprites, mob skins etc. A list of these can be found here. Modifying these is now supported since the Minecraft Version Alpha 1.2.2 Update on November 10, 2010.

It is worth noting that Pigmen, Zombie, and Zombie Pigmen mobs can use typical player skins (and vice-versa).

Changing Player Skins
A player can only change their character's skin if they purchased Minecraft. This is done on the Profile page by uploading a valid .png image file, which will then replace the default skin. The char.png file in minecraft.jar can also be changed and replaced, but then, only in that game the player character will appear different, and that is also the case for other people who use the default skin.

Creating a skin
A custom skin is a great way to personalize your player model and can be done either by using a variety of community-made skin editors or by editing Minecraft's char.png file manually. This is stored in the minecraft.jar file and can only be accessed by using an archive decompiler (such as Winzip, Winrar, or 7-Zip).

In Windows, minecraft.jar is located in "C:\Users\User\AppData\Roaming\.minecraft\bin", accessed by pressing + and entering "%appdata%" in the command box. This is done differently on a Mac.

Skin files must be 64 x 32 pixels.

Note: As of the 1.9 pre-releases and 1.0.0, all "bottom" textures (including hand and foot) have been flipped 180 degrees.

Capes
Capes (previously known as cloaks) are vanity items and are worn in addition to the player's skin. There is currently no way to obtain a cape on one's own, however there are various mods that allow players to do this.

Forum Discussion: http://www.minecraftforum.net/topic/648202-all-existing-cloaks-in-minecraft/

History
In October 2011, Notch renamed the cloak to the cape mentioning that the name was already wrong in the first place.

So far the Christmas and New Years Capes have only been given out in 2010 and not subsequent years.