An emulator is a program file you can place on your computer that allows you to play video games. It's essentially a simulator of consoles like the NES, SNES, Sega Genesis, Game Boy Advance, etc.
They allow you to download games (Called Roms) and play them on your computer. They can be downloaded on the internet, but you must do so at your own risk, as there is always the chance of unsafe websites carrying viruses and all that fun stuff.