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. >THERE ARE NO FUNCTIONAL 3DS, VITA, (PS3 RETAIL GAMES), XBOX 360, OR WII U EMULATORS
Contents
DOS
NES/Famicon
General: Nestopia
Accuracy: puNES or Nintedulator
Gameboy/Color
Gambatte
SNES/Super Famicon
General: Snes9x
Accuracy: Higan (formerly bsnes)
Master System/Genesis
General: Kega Fusion
Accuracy: Genesis Plus GX
PC Engine/Turbografx-16
General: Mednafen
Accuracy: Mednafen
Playstation
General: PCSX-Reloaded
Sega Saturn
Nintendo 64
General: Mupen64Plus
Accuracy: CEN64 (Note, it's early in development and won't be useful for quite some time)
Really N64 emulation is a mess, I'd actually recommend checking out this site and using the appropriate emulator/plugin per game.
GameBoy Advance
General: VisualBoy Advance-M
DreamCast
GameCube/Wii
Playstation 2
NDS
PSP
General: jPCSP
Up-and-coming: PPSSPP
For most audio, you'll need some method of decoding ATRAC3+. FFMPEG plans to cover that, but for now you need SonicStage.