THERE ARE NO FUNCTIONAL VITA, (SOME PS3, XBOX 360, WII U and 3DS RETAIL GAMES), PS4 OR XBONE EMULATORS
Accurate emulators might be less user friendly, or be difficult to set up. For that reason, there are two recommendations, one for more general use, and another one for accuracy.
For a list of emulators from every system, go here.
Multi-System
DOS
MSX
Famicom (NES)
General: Nestopia
Accuracy: Mesen, puNES or Nintedulator
Game Boy/Color
Arcade and Neo Geo
General: Final Burn Alpha
Accuracy: MAME
Super Famicom (SNES)
General: Snes9x
Accuracy: higan (formerly bsnes)
Master System/Genesis
General: Kega Fusion
Accuracy: Exodus
PC Engine (TurboGrafx-16)
General: Ootake
Accuracy: Mednafen
PlayStation
General: PCSX-Reloaded
Sega Saturn
Nintendo 64
General: Project64
Accuracy: Mupen64Plus
Game Boy Advance
Dreamcast
GameCube/Wii
PlayStation 2
Nintendo DS
PSP
PlayStation 3
RPCS3 (Highly experimental)
Xbox 360
Xenia (Highly experimental)
3DS
Citra (Highly experimental)
WiiU
Cemu (Highly experimental)