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
DOS
NES/Famicon
General: Nestopia
Accuracy: puNES or Nintedulator
Gameboy/Color
Gambatte
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)
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.