An emulator is a piece of software that uses computing power to emulate the hardware of another device, such as a video game console or a personal computer. As a general guideline, the Wii is capable of emulating most popular home consoles up to around the fifth generation (the N64/PS1 era).
The Wii has hardware that allows for native backwards compatibility with the GameCube, meaning such software can run at full speed as they would on the original hardware. While this is not actually emulation, methods that use this hardware will be included here due to the ubiquity of the term.
Virtual console
The Wii has many games from older consoles previously released on the Wii Shop Channel as Virtual Console titles. These include the Nintendo Entertainment System, Super Nintendo Entertainment System, Nintendo 64, TurboGrafx-16/CD, Sega Master System, Sega Genesis, Neo Geo, Commodore 64, and a selection of arcade titles.
For platforms such as the Nintendo 64, these Virtual Console titles may provide the best experience to play certain games on the Wii, as oftentimes the channels were built around of running a specific game.
Besides using homebrew emulators, you may also create Virtual Console "injects", where you take game files and inject them into a Virtual Console title in order to run that game with the provided emulation. A compatibility list for different injects can be found here.
Home video game consoles
NES/Famicom (Disk System)
Possible through FCE Ultra GX. Alternatives are FCE Ultra RX and FCEUX TX.
SNES/Super Famicom
Possible through Snes9x GX. Alternatives are Snes9x RX and Snes9x TX.
Nintendo 64
Possible through Not64, this is the recommended option as it is still maintained and updated today. Another option is Wii64.
Nintendo GameCube
GameCube games can be played natively through Nintendont, a tool that lets you play GameCube backups from an SD card or USB drive. All Wiis can play GameCube games, but the Wii Family Edition (RVL-101) & Wii Mini (RVL-201) cannot read GameCube discs, requiring you have a GameCube or an original Wii (RVL-001) to dump GameCube games. They also do not have GameCube ports or memory card slots, so you must either use a USB HID controller or solder on the ports yourself.
Sega Genesis / CD / 32X / Master System / Game Gear / SG-1000
Possible through Genesis Plus GX, except for 32X emulation which it does not support. Instead, use the PicoDrive core for RetroArch Wii.
Sony PlayStation
Barely possible through WiiStation or WiiSX. Compatibility lists for both WiiStation and WiiSX can be found.
Arcade cabinets
You can play most arcade games via the MAME-Wii emulator.
Handheld video game consoles
Nintendo Game Boy Advance/Game Boy Color/Game Boy
Posssible through mGBA, this is the recommended option and will provide the highest speed and compatibility. Another option is Visual Boy Advance GX.
Nintendo DS
Barely possible through DeSmuME Wii. A partial compatibility list can be found here; most games are incompatible or run very slowly.
Bandai WonderSwan
Possible through WiiMednafen, a multi-console emulator.
Home computers
DOS / Win9x
Possible through DOSBox Wii. Windows 3.1 and 95 technically work, albeit slowly; this emulator is much more practical for older DOS games.
Commodore 64
Possible through Frodo.
Multi system emulators
Retroarch
RetroArch Wii is a port of the RetroArch emulator. The Wii version supports a large number of consoles (to varying degrees of success).
WiiMednafen
WiiMednafen is a port of the Mednafen emulator that was originally intended to port a small selection of video game consoles, but ultimately resulted in the entire project being ported over. The emulator supports a total of 12 consoles:
- Nintendo Game Boy/Game Boy Color
- Nintendo Game Boy Advance
- NES/Famicom
- Virtual Boy
- Sega Master System
- Sega Game Gear
- Sega Genesis/Mega Drive
- Atari Lynx
- PC Engine/TurboGrafx-16/CD/Supergrafx
- NEC PC-FX
- Neo Geo Pocket
- Bandai WonderSwan
Emulators that don't exist
These are consoles or devices that will not work on the Wii, usually due to computing power requirements.
- PlayStation 2 and newer
- PlayStation Portable and Vita
- All Xbox consoles
- Sega Dreamcast
- Windows NT (XP/Vista/7/etc)
- Nintendo 3DS
- Wii U
- Nintendo Switch
Other consoles and further reading
For other consoles that have not been listed here, as well as other emulators, this wiki page from the Emulation General Wiki is a good source to start with.