There are basically 2 types of RJ45 Ethernet cable: standard and crossover. Standard is what connects a device to a hub, switch or router; crossover is what connects one device to another device.
My understanding is that the Wii USB-Ethernet adapter comes with a standard Ethernet cable. So if you're going to be connecting it directly to a hub/switch you're fine, but if you're going to connect it to your PC's network card (so your PC can route between your Internet connection and the Wii), then you'll need a crossover cable.
You can tell which is which by placing the plugs side by side. A standard cable will have the same colours in the same positions on both connectors; a crossover has a different layout (the wire colours will look different).