The Wii can't use an external HDD. The only one of this current generation of consoles is a PS3 (although I remember hearing somewhere about 360's and laptop hard drives).
The Wii can use SD cards however. If you're downloading a lot of games or looking for memory expansion, the Wii can save and read directly from the SD card. Not sure, but I think there's an upper limit to the amount of space on an SD card you can use, but I think you'll have to research that to get a reliable answer.