Bassicaly, Emulation is the process of simulating the hardware of one machine on another machine so that the second machine is able to run software designed for the first machine ( example: gba on the pc )
The most popular type of emulation is video game emulation. This is when someone plays games from NES, SNES, Genesis, etc. on their PC. Some websites provide EMULATORS (software that allow you to run games, designed for other systems, on your PC) and ROM FILES (games you run using an emulator) for download so you can re-live all those classic games you enjoyed since you were a kid or something.
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It's not illegal if you originally own the system and game. If you've never bought the console, or the game you want to play on it, THEN it's considered illegal because then you're cheating, finding a way to play free games.