nono I'm not talking about configuring a new Xbox 360 with your existing gamertag. All you have to do is enter in your gamertag, I know that. I'll explain how this works a little more.
Basically, Microsoft does this as to curb abuse. Basically, whatever Xbox you use to sign up for the Xbox Live membership, it ties that gamertag to that Xbox.
Now you can use your gamertag on anyone's xbox, and as long as you are online, it functions exactly the same, except 1 big difference. When you actually go offline and you are not connected to the internet, on the xbox that is tied to your gamertag, you can still play your xbox live arcade games. On any other xbox, you need to be connected to xbox live to play.