How do you change your character from one server to another in World of Warcraft

First you go to your account management section of Battle net. Then you click on the World of Warcraft game. Underneath the license and download areas there is the character re-customization area. This is where you go to change servers, faction, names.

Quoted from the paid character transfer page on the world of warcraft website:

• The paid character transfer is not immediate and may take several days to complete. Your character will not be available for play during this time.
• Each character you transfer costs $25.00 USD . This covers the character's ONE-WAY TRANSFER to a new realm, a new account, or both at the same time.
• Your character will not have any guild or arena team affiliation upon arrival at the destination realm.
• Each character may only transfer once every 3 days.
• You will need to rename your character if his/her name is taken on your destination realm.
• Character transfers between accounts are subject to restrictions. Click here to learn more.
• Characters who undergo a Paid Character Transfer are ineligible for Realm First Achievements for two months after the date of the transfer.