From Diablo Wiki
Transmogrification (Transmog) is a system coming to Diablo 3 in the expansion Reaper of Souls. It's taken directly from the popular system of the same name in World of Warcraft, and enables characters to change the appearance (but not the function) of their gear. The Mystic NPC Artisan is the place to go to transmog in Diablo 3 Reaper of Souls.
Transmog will function entirely via the Mystic NPC, and it will work much like the Jeweler and Blacksmith work; some recipes (for different transmog looks) will be enabled by training the Mystic to higher levels, and other transmog appearances will be enabled by finding items and identifying Legendary and Set items. There might even be special, limited-edition transmog appearances given away in contests, in Blizzcon goodie bags, etc.
- Place item you wish to alter into the Mystic's Transmogrify interface.
- A list of possible looks will display as well as a portrait of your character.
- Click item to view it on your character.
- Click Transmogrify once you find an appearance you wish to change the item to.
Scope and Restrictions
- Level up the Mystic for new appearances and identify Legendaries and Set items for additional appearances.
- Legendaries and Set items found before the expansion (referred to as legacy items) will not unlock the ability to transmogrify items into their appearance.
- Items will only unlock their appearances in the mode (softcore/hardcore) they are found in.
- Items transmogrified will not bind to the player's account so they remain tradeable, unless they were originally bound to the account, i.e it's been enchanted.
- If an item has been transmogrified and can be traded it will retain its appearance once traded, even if the recipient hasn't unlocked that appearance with their own Mystic.
- If you trade for an item that has been transmogrified and you do not like the appearance you can pay your Mystic to remove it. If you remove the appearance from the item and you haven't unlocked the appearance yourself you won't be able to re-apply it via your Mystic until you have unlocked it.
- Fist weapons are in their own category for transmogrification, and can't be changed into one-handed swords or other weapons.
- Wands are in their own category too,  as are Flails. 
- Ceremonial knives overlap with daggers and some spears so can be transmogrified into those.
Development of Transmogrification
Grimiku: We know inventory space is precious, so you won’t be required to keep the “visual” item in order to use it’s appearance. Instead, you’ll unlock the item’s appearance across all Normal (or Hardcore) characters on the account once you’ve picked it up. Any item appearance that’s been unlocked will be stored in the transmogrification UI.
- Reintroducing the Mystic - CM, Blizzard Entertainment 24/10/13
- Trading Transmogrified items - CM, Blizzard Entertainment, 15/11/13
- Transmog Categories - CM, Blizzard Entertainment, 01/11/13
- Flail's category - CM, Blizzard Entertainment, 02/02/14
- Acquiring 'plans' to access - CM, Blizzard Entertainment, 10/09/13
- Blizzard Defends Transmogrification - CM, Blizzard Entertainment, 01/11/13