Wii/DSi shop, Internet channels, and the like will not be affected
[Update: Nintendo of America, as well as Australia, have also announced the end of the Wi-Fi Connection service. Looks like it's across the board.]
Nintendo of Japan has announced that they're ending all DS and Wii online services for games that took advantage of the capability. You'll still be able to access things like the DSi and Wii Shop channels, Hulu, YouTube, and access the internet via the respective browsers. You just won't be able to play any games online with others anymore.
The Nintendo website has a giant list of what games will lose their online access. Games such as Super Smash Bros. Brawl, Wii Music, Mario Kart Wii, Animal Crossing: Wild World, Tetris DS, Pokemon Pearl, Pokemon Black/White, and Metroid Prime Hunters, are just some of the higher profile titles Nintendo has laid out.
The Wi-Fi Connection service, which originally launched back in November 2005, will come to an end on May 20, 2014. We've reached out to Nintendo of America to see if this is just Japan, or if it's a worldwide discontinuation.