The original Xbox wireless adapter is only WEP-compatible (since this was the wireless security technology available at the time). If your network is set up to use WPA or WPA2, you can change it to WEP by consulting your router's manual (or by Googling -- because who keeps the manual?)
If you aren't sure whether your network is using WEP, log into your router control panel (Google your router's manufacturer for instructions on doing this).
Using the supplied AC adapter and ethernet cable, connect the wireless adapter to your Xbox 360.
On your Xbox, navigate to Settings > System > Network Settings and select your wireless network. Enter your network's WEP key.
We've got a guide to help you find your network security key, should you need to find it again.
If your Xbox states that "The given WEP key is invalid", first try turning your Xbox off, unplugging the wireless adapter, and plugging it back in. If this still doesn't work, try entering the WEP key generated by your router rather than the WEP passphrase you created. For example, if you set your WEP passphrase to "killallhumans" in your router's administration panel, rather than entering "killallhumans" in the Xbox wireless settings screen, enter the encrypted key that your router generated on the same page (i.e. "f62nji910ena1p6x8c").
Sit back and watch something excellent on Netflix. If you're a fan of howchoo, I recommend MacGyver.