Basically with a modified Zelda: Twilight Princess save some hackers have managed to run unsigned code on the Wii (just a few lines of code right now).  The implications though could mean that this might become the Wii's equivalent of the PSP's GTA exploit.  The good news is that running games without requiring a modchip likely wouldn't be a major problem for a console that has a large installed base like the Wii.

http://www.tehskeen.com/

nice but this was expected to happen eventually

That means WiiLinux in a short time. I think thats the most important implication.

Won't Nintendo just patch this in their next system update?

Highly likely they'll block it with an update, like they blocked that GameCube region-free thing. And they'll put the update in Brawl and Fit, so you have to get it.

I can't think of anyone who has a Wii that doesn't want at least one of those =P  So yes that would very likely solve the problem for the vast majority of Wiis.  Especially if they sneak it in with future updates that feature other goodies people want (but for free).

It really depends on how the exploit works. My impression is the exploit is within Zelda itself (otherwise, why not mention it works with other games too). It appears that it requests the save data, and then once this is decrypted and loaded into memory, it is executed. Is Nintendo able to patch games? If not, is there a way they can modify the loading mechanism without causing issues in current games?

Also, given that this allows for code to be executed in Wii mode, it may not be long until custom firmwares are available just like the PSP.