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There is no legal way of playing NES, SNES, N64, or GameBoy games on your computer though, unless you own some way of backing up your own cart.
Actually, I think that's what Rick was just talking about. Most people know that distributing ROMs is illegal. But even making a "backup" in the first place is what companies like Nintendo have been trying to fight (and I'll take Rick's word on it that it's illegal).
Anyways, considering that KnowsNothing apparently knows nothing about emulators, and would like to know everything about them, I'll try my hand at a basic, yet in-depth explanation.
ROMs:
Cartrige-based videogames are programs built onto ROM (read-only memory) microchips. You plug them into the dedicated machine that they're built for, and they add their program to the videogame machine, making it do stuff. A long time ago, people figured out how to stick another kind of machine (usually called a Game Copier) between the cart and the system to read exactly what it is that the cart's doing to the system. Then it can copy a "ROM image" onto something like a floppy disk. You can take the cartrige out of the copier, and use the "ROM image" from the floppy disk instead, and the copier should be able to fool the system into thinking it's really the game cartrige. Since this "ROM image" is now on a floppy disk, you can easily give it anyone else with a game copier, or copy it to your PC, where you can then give it to almost anyone on the internet.
This practice is ILLEGAL. It doesn't mean that it doesn't happen, it doesn't mean you're not allowed to know it happens (although PGC has the right to make their own rules about that kind of thing while we're here), it just means you can go to jail if you try doing it yourself.
Translation Patches:
One of the cooler things about ROMs, and a major temptation to break the law and use them, is the fact that people can make "patches" for ROMs. Devoted fans of a foreign game, frustrated by seeing it get passed up for localization, can look inside the contents of a ROM image and (after a lot of effort) find where all the foreign text is hidden. They can then go about translating it themselves, and creating a patch for the ROM image. If you put the patch onto a ROM image, it will replace certain sections of the image with their translated replacement text. Just using an "Import Walkthough" to play a 100% legit version of the game is way more legal, but there's no denying that it pales in comparison to the smoothness of a well-made translation patch.
The patches themselves are supposed to be legal (although there is no legal use for them), even though you just know that
somebody has to have broken the law at least a few time for one of them to have been made.
Emulators:
An emulator is (like Rick said) a computer that's pretending to be another computer. Most are legal. Some aren't. Like I know that most emulators for the MSX (the Japanese personal computer where games like Metal Gear first appeared) require a copy of the MSX's BIOS (Binary Operating System) in order to run properly. That's illegal. But when people take the time to write their own operating systems for their emulators, and spend years working out all the bugs and flaws, then that's fine.
There are emulators for the PC out there that emulate the NES and SNES, and while it seems that they have no possible legal use (since you can't really stuff an NES cart into your PC), there are some "homemade" games being developed for them. Those are legal (unless they start copying massive sections of other people's game programs to build their games). And there aren't really any laws that say the things you make have to be useful (as far as I know). So the emulators themselves are legal, but they just keep getting used for illegal purposes.
And then there are emulators out there that emulate systems like the Sega CD and the TurboGrafx CD and the Playstation. These are really cool, and actually useful, because you can put the game's CD into your CD drive and play the real games on your computer. That doesn't stop people from being able to make "Disc Images" and trading them over the internet, or playing bootleg games in the emulators, but illegal bootlegging was already a problem with these systems, so it's not really the fault of the emulators.
Official Emulators:
There are games like Final Fantasy 4 to 6 and Chrono Trigger on the PSX that actually use SNES emulators on the Playstation. The version of Metroid included as a connectivity bonus in Metroid Prime runs on the GameCube in an NES emulator. Because even though the GameCube was made by Nintendo, and is way more powerful than an NES, it just isn't an NES. I don't think that most of us know all that much about this kind of emulation. It's designed to be the kind of "magic trick" that you don't even see. When it does it's job right, you shouldn't even know it's there.