Answer:
In World of Warcraft, to transfer from one realm to another you have to log into your account management page at the Battlenet site, click on manage games, click the world of warcraft link, then you might have to scroll down a tiny bit and select the character transfer option. Follow the steps through. You will be charged $25 for the transfer and you can only use a credit card. The transfer usually takes about an hour, so dont play the character otherwise it will take longer.
Be sure to read the Character transfer FAQ