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You need to connect your Xbox to your modem/router and have subscribed to Xbox Live. A grey ethernet cable comes with your Xbox which has to be plugged into the back of your console and then into the back of your modem/router.

After that enter the dashboard of your Xbox (just start it up without a disc in the disc-drive) and go to the Xbox Live tab. Follow the steps there and subscribe for a (free) Silver membership. A Gold memebership is required to play online. You can buy them either online or at game stores, but you can also claim free trials that come with new games. For example, GTA IV gives a one month free trial that comes along with the game information booklet, as a code printed on a 'steps-to-take' booklet.