The wireless cable modem... the thing that connects the computer to the internet...
This is how people end up making threads like this, because people who have no clue what they're talking about try and answer questions.
Assuming you have a high speed connection of some variety, either DSL, satellite or cable (and if you don't, leave now, because dial-up is incompatible with a wireless router), you have a high-speed modem that connects to your computer via an Ethernet cable, or in some cases, a USB cable. A high speed modem is NOT a router, and unless you specifically purchase a high speed modem with wireless router capability built in, which are available, your modem won't act as one.
A wireless router is a piece of equipment that goes between your modem and your computer, and connects to both via Ethernet cables. The router takes your high speed Internet connection and converts it into a wireless signal that it broadcasts over the router's effective range, then passes the Internet connection through to your computer. The wireless signal is what your Wii, DS, laptop, Blackberry, whatever, acquires in order to access the Internet wirelessly. A good wireless router can be found for $40 if you look in the right place, only a few dollars more than Nintendo's USB Wi-Fi connector and infinitely more useful.
If you want to get online wirelessly with your Wii or DS, you'll need either a wireless router or the USB Wi-Fi connector, and that's all; the Wii and the DS are both wireless-capable out of the box. Dial-up users need not apply.