So the arcade Triforce system is basically a modified GameCube that plays games on Sega GD-Roms (the same type of discs Dreamcast used), and I think custom hard drives. Called Triforce because of Zelda, of course, but also because of the three companies that designed it: Nintendo, Sega, and Namco. Three of the games made for it are F-Zero GX, and Mario Kart Arcade GP 1 and 2. Because Triforce is so similar to GameCube, the games are playable on Wii/ Wii U (both also containing modified GameCube hardware), although an actual GameCube is not quite powerful enough, due to the added hardware of the Triforce unit.
Nintendont (no apostrophe) is software for Wii and Wii U that allows GameCube backups to be playable on SD cards,, USB drives or system memory. A side effect, I guess, was allowing Triforce arcade games to play. Programmers were able to make the games play virtually identical as they did in the arcades. Because Wii and Wii U contain GameCube hardware, this is NOT emulation, just a "key" to get the games to run, basically. Triforce will always be closer to the actual original arcade experience, since the systems are so similar. In fact, only older versions of Wii/ GameCube emulator for PC, Dolphin, even have Triforce compatibility. Not sure why the Dolphin developers don't seem to care about Triforce support.