Availability: 100% Free, Homepage

This is a mass system emulator with support for pretty much everything: PlayStation 1, N64, Nintendo DS, SNES, NES, Game Boy, Game Boy Advance, NeoGeo Pocket Color, Virtual Boy, Genesis/Mega Drive, Sega CD, 32X, Sega Master System, Game Gear, PC Engine/TurboGrafx16 (plus CD support), Wonderswan Color, Atari Lynx, arcade (MAME), and more. It offers lots of features: screen filters (shaders), save states, different screen overlays, cheats, netplay, and more! It's highly customizable! I highly recommend RetroArch.
*RetroArch is also available for iPads and Apple TV!
*Need help? I don't have a specific tutorial for RetroArch on iOS, but you can check out my RetroArch for Android tutorial. Both versions are nearly identical.

Availability: 100% Free, Homepage

Provenance is a multi-system emulator that supports the same amount of video game systems as RetroArch (which is pretty much everything). As you'd imagine, that makes them close competitors. Where Provenance shines over RetroArch is that it has a more visual UI. The UI also more closely follows that of iOS. Its Nintendo 3DS emulation is good as well. However, Provenance doesn't let you pick from a variety of different cores like RetroArch does.
*Provenance is also available for iPads!

Availability: 100% Free, Homepage

This is a mass system emulator with support for NES, SNES, N64, Game Boy/Color, Game Boy Advance, Nintendo DS, and Sega Genesis. Features include save states, cheats, gamepad support, and beautiful overlay skins for the onscreen buttons. You can sync your games, saves, and skins with Google Drive or Dropbox.

Availability: Free with ads

A simple, good PlayStation 1 emulator that offers save states, custom controller skins, and cover artwork for titles. The only annoying thing about it is that you're presented with fullscreen ads as you use it. If that annoys you, can you use RetroArch for PlayStation 1 emulation.

Availability: 100% Free, Homepage

The iOS port of the #1 open source Sony PSP emulator! It's a superb emulator with many features. PSP is rather CPU-heavy to emulate, so don't expect games to run perfectly on your iPhone. Fortunately, PPSSPP offers lots of options to help you optimize performance to help games run smoother.

The most popular emulator on OSX. It's a multi-system emulator that supports around 30 systems! It has a nice and clean UI, too; as expected from the reputation of Mac apps. Make sure to check out its wiki for help and to explore all the features available. Don't forget that several systems such as PlayStation 1 and Sega CD require the BIOS files to work, which you can download below in Extras.

If you're not happy with OpenEmu or prefer standalone emulators, you can download all of them from the #1 website for video game emulation on OSX, MacScene.net! It has a pretty active community, too. • You might also want to check Bannister.org because that guy alone made the majority of the available video game emulators for Mac. In other words, he's like the father of video game emulation on OSX. • For some reason, MacScene.net doesn't have a PS1 emulator posted. The best PS1 emulator on Mac is DuckStation.