From Diablo Wiki
Transmogrification (Transmog) is a system that allows characters in-game to change the appearance (but not the function) of their weapons and armor. It's a purely aesthetic device, akin to Armor Dye but changing the graphic, not just the color.
To unlock different Transmog looks, players must level up the Mystic. As she levels up she will constantly gain new transmog options for every item slot, until at level 12 she can make any item look like any other item tier. To grain transmog looks of legendary or set items, they must be found. When a player finds a particular legendary item for the first time, a "Transmog Unlocked" text pops up.
The transmog system is common in other RPGs, and the way it works in Diablo 3 is more or less directly taken from World of Warcraft.
How to Transmog
- 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.
- Prices are generally 500 gold for any normal look and 50,000 gold for a legendary item graphic.
- 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
What items can be transmogged into which others has changed repeatedly since launch, with more freedom and options added over time. There remain some inconsistencies, such as Crusaders able to transmog 1H flails into any other 1H melee weapon, while they can not transmog 2H flails into any other items (though 1H and 2H flails look basically identical when equipped).
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.
Transmog Upgrades in Patch 2.1
I can tell you up front that you won’t be able to Transmogrify your gear to look like another class, but you’ll have more options for your own class-specific gear when all is said and done.
But what if you want to turn a normal shield into a Crusader shield?
This should work as long as you do it while playing your Crusader. The shield will default back to its normal appearance if you transfer it to another character, though.
We especially need more options for 2H flails. 1H flails can be 1H swords/axes/maces, so why can’t 2H flails be 2H swords/axes/maces?
The current restrictions on weapon transmogrification will still be in place after this change, but I’ll bring this specific situation up at our next meeting.
That's good to hear. I will assume my barb can finally transmog Raekors on her gear (also picked this up when I caught part of yesterday’s stream, they said someone had this working internally).
This is a good example of what will be possible once these changes take place. Currently, players are unable to Transmog gear into the Raekor set, and even in a situation where the original piece of gear in question is a piece of Raekor. However, after the change a Barbarian could alter their gear to look like Raekors, so long as itâ€™s been unlocked as an option. The item would default to its original appearance when transferred to a different class.
Please for the love all all things holy let us transmog our 2hand flails to other weapon types.
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