It was only in the last few months that homebrew really started to go anywhere. One exploit that's currently being pursued is a save-game hack with Zelda: Twilight Princess. The advantage to this is that no modchip would be necessary and the Wii wouldn't have to be opened up to run homebrew. The guys working on the exploit have released a very barebone loader, but it's more a proof of concept; as of now it doesn't actually load anything, but according to them a full ELF loader is coming soon.
As far as apps, nothing's really available because up until recently homebrew wasn't even possible. Give it another year or so, though, and I'm sure we'll start to see some good software for Wii. I'm personally waiting for Wii media center that supports streaming media over WiFi and playback of DVDs.
Check out wiibrew.org.