Of course, there's no key you need to enter or anything. You just need your account name and password, as alyways
The only thing is the DREADFUL and LONG installation process! Not only the installation, but downloading and installing all the patches... It can take around 6 hours if you're downloading everything! (Yes, it happened to me, luckily i burned all the patches up till now on a DVD so I don't have to worry about downloading them again)
What I suggest, especially if you don't want to download the gigabytes of patches and to have tp set up your add-ons again, is to COPY the WOW folder from the PC you're using it on to the new one. Burn it on few DVDs or a USB stick or something and just copy and paste it. It's perfectly safe and functional since the installation doesn't leave any registry settings or similar.