Guide to Transmogrification 1: The Basics
Imagine having a warm winter coat. It's cold outside so you want to wear it, but it doesn't match anything else in your wardrobe, and it doesn't flatter your figure. Then imagine being able to make it look like a thin jacket or a nice blazer instead, but still being warm. Nice, right?
Transmogrification is the act of giving a piece of armour or a weapon in World of Warcraft the look of another item while it retains its original properties.
 Now, before you get too excited, there are limitations. Your warlock can't transmogrify (also called transmog, xmog or just mog) her robe to a plate chestpiece, and your druid can't transmog his staff into a rogue's dagger.
Here are some basic guidelines:
1) You need to have both items (the one you want transmogged and the one you want it transmogged to).
2) You can't transmog items of grey and white quality or legendaries (but there are rumours that the latter is subject to change).
3) The desired item has to be of a level that your character can equip (so a level 30 character can't transmog their gear into items only available to level 80+ characters).
4) You can only transmog cloth to cloth, leather to leather, mail to mail and plate to plate.
5) When it comes to weapons, the same thing applies. A two-hand weapon can't be transmogged to a one-hand weapon. (Bows, crossbows and guns can, however, be transmogged to eachother.)
6) If an item is class specific and your character is not that class, you can't transmog to it. So your hunter can't get the look of a "shaman only" item.
More in-depth rules can be found on WoW Roleplay Gear or World of Warcraft Transmogrification Guide.
So where do you bring your items to have them transmogged? There are four so-called warpweavers (see what I did there?) in World of Warcraft who can transmog your items. Two of them are located in Alliance cities and two of them in Horde cities. Wowhead can tell you where exactly to find them.
(Transmogging in Orgrimmar. Just drag the items you want transmogged to from your bags unto the right spot. Or right click on them.)