Lakka is an OpenELEC/ RetroArch-based Linux distribution which transforms small computers into retro games consoles.
There's support for emulating the Atari 2600, Lynx, Game Boy/ Colour, NES, SuperNES, Sega 32X/ CD, Game Boy Advance, PlayStation, PSP, Nintendo 64/ DS, PSP and many more.
You can do any of this on a PC by downloading the appropriate image and writing it to a USB key, but Lakka also at least partially supports various small computers: Raspberry Pi/ Pi2 (VideoCore), Allwinner 20 : Cubieboard2, Cubietruck and Banana Pi (Mali-400), i.MX6: WandBoard, Hummingboard and Cubox-i (Vivante), and Amlogic: Odroid-C1 (Mali-450). (Note that not all computers can emulate every console. The developer's Support page tells you more.)
There's direct support for a range of controllers, and Lakka can be configured to get others working, with a little effort. (The developer lists the best three as "XBox 360 wired controller", "XBox 360 wireless controller + its Microsoft adapter", "Wired Dualshock 3 controller".)
Lakka won't somehow give you instant access for every game for each console. It's just another emulation platform, and you'll need to locate ROMs for yourself, just as with any other. But the developer does provide a good set of instructions to help you get started, and if you've used an emulator before then you'll soon be at home.