Cape

Capes (previously known as cloaks) are extremely rare vanity items and are worn in addition to the player's skin.

Obtaining
There is currently no way to obtain a cape on one's own on Java Edition, although there are various mods that allow players to obtain one, only visible to that player (or in some cases, other users of that mod).

In the Legacy Console Edition and Bedrock Edition of the game, some skins from various skin packs include capes.

Players on Java Edition who legitimately have one or more capes can choose which to wear (including no cape) by going to their Minecraft profile.

Players on Bedrock Edition can equip or unequip capes for their skin via the Character Creator, and on an older version you could make skin packs with your own custom capes.

Server Capes
Server capes are capes used exclusively on Minecraft servers. They can either be used as a reward, or they could be given to specific people.

Unused Capes
MINECON 2011

These capes were featured in a poll where MINECON 2011 attendees could vote for their cape of choice. The red creeper cape won MINECON 2011, and the blue cape with the pickaxe was for MINECON 2012. These following capes are still unused though may appear in the future.

Elytra
When a player is using a cape and has equipped an elytra, instead of overlapping the elytra, the cape gets hidden and a custom design of the elytra shows on the wings depending on which cape the player has on.