You can download a TON of games for free from XBox Live. Here is how to do it - and how to download for-pay games too.
First, use the central X button to log into your XBox Live enabled account. If you haven't done this yet, then read my other instructions on how to set up an XBox Live account.
Once you're logged in, go to the XBox Live tab (the leftmost tab) on your main dashboard. Choose the XBox Live Marketplace.
Go to Game Downloads, then choose the "All Games" tab. Go down to XBox Live Arcade.
You'll see categories here of all the cames they have. Let's say you're interested in the original Gauntlet game, with the Green Elf Needs Food Badly. That would be under "coin-op classics". You can also see Joust, RObotron: 2084 and Smash TV in here. Choose Gauntlet.
Now you see your options. You can get the free version which is a trial version. You can also pay 400 points to get the full version. When you choose what you want to download, it'll take a few minutes to move it to your local hard drive. Once you get it you can play it whenever you wish, even when you're not online!