You only need the xbox wireless adapter if you plan to use it wirelessly. Otherwise, just connect them with a simple ethernet cable. Xbox wireless adapters are pricey. I used to own xbox 360 before I switched to ps3. A cheaper alternative may be a wireless game adapter.
P.S. Opening your NAT for your connection would benefit gaming. Do this by logging into router > Firewall settings > DMZ Host > type IP address of xbox 360 (not of computer or router, but make sure it is the xbox 360's IP). If further assistance is needed, feel free to reply with questions.
I have my 360 wired to my modem. It saves you money from having to buy the $100 wireless adaptor and gives you the best connection. I can sacrifice a slightly slower download speed on my wireless laptop, but I'm not willing to sacrifice a slower connection on my 360, especially playing MW2.