Recommended emulators
All links here will take you to the latest build of that emulator, if you what more info regarding the emulators, either check the Emulators page, or use Google.
Contents
Nintendo
NES/Famicon
General: Nestopia
Accuracy: puNES or Nintedulator
SNES/Super Famicon
General: Snes9x
Accuracy: higan/bsnes
(I'd recommend using bsnes core in RetroArch to get around some less-user friendly implementations of bsnes)
Nintendo 64
Accuracy: Mupen64plus
Really N64 emulation is a mess, I'd actually recommend checking out this site and using the appropriate emulator/plugin per game.
GameCube/Wii
GameBoy/Color/Advance
General: VisualBoy Advance-M
Accuracy:Gambatte (only emulates GameBoy and GameBoy Color)
NDS
Sony
Playstation
General: PCSX-Reloaded
Accuracy: Xebra or Mednafen
(Scroll to the bottom of the page and click on the image for Xebra)
Playstation 2
PSP
General: jPCSP
Up-and-coming: PPSSPP