A router is what broadcasts the wireless signal from your modem (where the internet is connected from your wall usually, into your computers).
You need a router or one of those Nintendo adapter things (which I've heard are crap) to access the Nintendo wi-fi.

Alternatively, I think for platinum, they made another trading option so that you can simply connect locally (without wifi, simply from DS to DS), but I'm not sure...I've never gotten to try it.