whats not to understand? A router simply splits the internet connection coming out of the modem into 4+ seperate lines so that you can hook up more than one device to the internet (i.e. PC, Gaming Console, Server, etc...) Now when you have a wireless router and a device with Wi-Fi (Wireless Fidelity) capabilities like a Laptop or the Wii, you can now connect to the internet without using any wires but instead a Wi-Fi signal.
but if you dont have a router you can use a adapter that plugs into a usb court on your pc and it tricks the computer into thinking its a access point and it shares the connection with the ds, and hopefully the wii without using any wires.
I only have dial-up at my home, for my online I'll have to goto someone else house probably, Nintendo has stated previously that dial-up would be sufficient but I doubt when the games are released they will even support dial-up.