Editors Note: This DOES require homebrew to run, mostly.
Emulators on 3DS are designed to run games from other consoles using custom firmware.
Diddy Kong Racing on a New 3DS XL.
Most emulators run older consoles, but some, like TWiLight Menu++ essentially turn the 3DS into a pimped up DSI. Another, known as New Super Ultimate Injector can disguise ROMs as Virtual Console files. It allows you to run NES, SNES, GB, GBC, GBA, Genesis, Game Gear, and PC-Engine ROMs.
VirtuaNES 3DS runs game ROMs from the Nintendo Entertainment System on your Nintendo 3DS.
SNES9x 3DS runs Super NES ROMs on the Nintendo 2DS/3DS.
Daedalus X64 3DS allows Nintendo 64 ROMs to run on a New Nintendo 2DS XL/ New Nintendo 3DS/New Nintendo 3DS XL.
PicoDrive for 3DS allows Sega Master System/Genesis/Sega CD ROMs to run on the Nintendo 3DS.
TemperPCE for 3DS allows for PC-Engine ROMs to run on the Nintendo 3DS.
Z26 3DS allows Atari 2600 ROMs to run on the Nintendo 2DS/3DS.
GameYob 3DS runs GB/GBC ROMs on your Nintendo 2DS/3DS.
mGBA runs GBA ROMs on your Nintendo 3DS.
r3Ddragon runs Virtual Boy ROMs on your Nintendo 2DS/3DS.
3DeSmume is a Nintendo DS loader for your Nintendo 3DS.
That definitely isn't all of them, but those are by far the best out there.