Dolphin
Dolphin Emulator is a GameCube emulator and Nintendo Wii emulator that started off its routes as just being a pure GameCube emulator. However once the release of the Nintendo Wii came round, the team realized that the hardware was so similar that increasing Dolphin’s support to also cover the Nintendo Wii was not a huge technical challenge. They found that it did not require significant rewrites of Dolphin Emulator to be able to also be a Wii Emulator.
Dolphin Emulator made its mark by being the first GameCube emulator that was actually able to successfully emulate a commercially released game. However despite achieving this, Dolphin was nearly gone completely when it was discontinued in 2004. Luckily a year later in 2005, the developers revived the Dolphin Emulator project adding minor improvements and beginning the first basic audio support.
Then in 2008 is when Dolphin Emulator really started making its breakthroughs after being moved into the public domain and being open sourced. Leading to its development as the best emulator for both the GameCube and the Wii. Dolphin emulator now successfully emulates almost every single game available on both systems. Its development has seemed to of never slowed down as the now large amount of developers have managed to successfully emulate even some of the most complicated functions of the GameCube and Wii.
Thanks to its large pool of developers, Dolphin Emulator is the best GameCube emulator about, and the best and only Nintendo Wii emulator. However expect with some games, to emulate them perfectly you will require a computer with decent hardware, as some functions are quite resource intensive to emulate.