Can you take your World of Warcraft account from one computer to another?

In a very few circumstances it's possible to transfer a character, but these are strictly regulated. One example is when a person had an account as a minor, and got his own account when he turned into an adult. Then it's possible to transfer characters from the old to the new account, since the old one would be in the parents' name.

Other than that it might be possible if all accounts are owned by you and placed under a single battle.net account.