The USB Connector is a fall-back plan to the Wireless Router. You can have one or the other, you don't need both. I, myself, have the USB Connector, and it plugs into your computer's USB 2.0 slot, it doesn't plug into a Wireless Router or whatever.
Although, I recommend you go buy the router. No stores carry the USB Connector for DS (or at least here they don't), and Nintendo can take up to 10-11 days to send you the Connector through mail.
Buy the router and you should have no problems. Just make sure to take down the Firewall to allow Wifi to connect to your internet. Most routers don't recognize Wifi as what it is ;x