To use the Xbox Live Arcade you have to have a broadband internet connection and an Xbox Live account (at least Silver). These games aren't free, unfortunately, and you have to redeem Microsoft Points to get them. They range from as low as 400 points to as high as 1200 (so far). The best part of the Xbox Live Arcade is that you can download demo versions of the games for free so you can try them out before you decide to buy them.
One major advantage of buying them (besides having them forever and being able to play the full game) is that each arcade game has achievements and points that add to your gamerscore. This gives you a little bit of extra incentive to keep playing and working at your high scores.
If your Xbox ever breaks down or you need to transfer games to a memory card or different hard drive, don't worry because you won't have to pay for them again. All of the downloads in the Xbox Live Marketplace and Xbox Live Arcade are tied to your gamertag so you just need to sign onto Xbox Live and you'll be able to download them again if you need to.
Thus far in the Xbox 360's life, the arcade is definitely proving to be one of the coolest new features.