There is the homebrew channel with bootmii/boot2 (if your Wii is old enough for this) and bootmii/IOS.
From there, there is the ability to play retail Wii games with a backup loader, as well as backup virtual console and wiiware games.
You can also load GameCube games with homebrew providing your Wii has the GameCube ports on it.
If you would prefer to load backups outside of homebrew, and especially for GameCube games (besides the ability to play GameCube games with audiostreaming) you can get a WODE. It is particularly useful if you would prefer to load backups without a homebrew and maintain the Wii as though it is being used with retail discs, in the sense that the Wii message log will reflect games being all loaded from the Wii's Disc Channel.
It is also possible to use a virtual NAND with a homebrew like neek2o, with the ability to load multiple (region) NANDs on-the-fly. This can make playing backup virtual console and wiiware games more convenient so that your real Wii's NAND does not get too many modifications. It is also convenient to load alternate region games without homebrew (like with a WODE).
Outside of playing backups on the real/virtual NAND, there are a lot of homebrews available.
There are homebrew games and applications, of which a lot of are on the wiibrew site (
www.wiibrew.org). These generally require them to be loaded from the Wii's NAND as opposed to a virtual NAND, as the virtual NAND has restricted access to read/write to the SD/USB, especially for homebrew.
There are emulators for different consoles, there are games, and there are media players. There are also other types of homebrew, like setting up the Wii as a server of sorts.. there are a variety of homebrews. I would recommend checking the wiibrew site for a list of homebrews available..
There have been competitions from different places to promote entering a homebrew for a prize, so there are also homebrews to choose from with that nature in mind.
The Wii is generally meant to play games. Besides homebrew and playing games/backups on a Wii, I can't see a purpose outside of those besides using the internet.
You can officially browse the internet and use official internet based channels..