Since the Nintendo Wii has a Opera-based browser, several sites list games that are playable on the Wii. Here are some of these sites:
These games are flash games that can be controlled with just the mouse, which the Wii remote emulates.
Currently there are no online multi-player Wii games. But, I was thinking that it would be pretty easy to create an online multi-player web game. You would play with other people that were connected to the web site at the same time. The only problem that I can see with this solution is that the game would only be able to use the mouse-like controls that work with the browser.