Dolphin is a emulator for the Wii and Gamecube. This is one of the most advanced emulators out today as it came emulate a Gamecube or Wii game very effectively with a high compatibility rating. This would be my pick for emulating Gamecube or Wii.
Need an emulator for the Playstation 2? With PCSX2 you have found your match. Keep in mind that while this emulator can run most PS2 games, it does require a fairly powerful computer to run smoothly.
PPSSPP is my pick for PlayStation Portable emulation. PPSSPP is an acronym that stands for "Playstation Portable Simulator Suitable For Playing Portably". PPSSPP was created by a Dolphin co-founder and can play a wide selection of PSP games.
DeSmuME is a Nintendo DS emulator that emulates both screens of this dual screen portable console. The bottom touch screen can be simulated with your mouse. This DS emulator has been developed for quite a while which means more compatible games. Note: If you use linux as your operating system you would need to self-compile DeSmuME.
While technically not a console emulator, DOSBox enables you to play classic DOS computer games. Who wouldn't want to play 'Day of the Tentacle' or 'Oregon Trail' again?
One of the best Playstation 1 emulators available, ePSXe supports emulating nearly all Playstation 1 games.
A list of emulators would not be complete without MAME. MAME stands for "Multiple Arcade Machine Emulator". MAME emulates arcade games and helps to preserve a piece of history.
Project64 is one of the best Nintendo 64 emulators available. While it may take some tweaking and extra configuration to get some games to work (most notably the sound plugin), Project64 still is very easy to use and supports most Nintendo 64 games.
Snes9X is arguably one of the best SNES emulators of all time. It is lightweight, powerful and compatible with every SNES game that I have ever tried. Definitely a solid choice.