From Sega Retro
An emulator is a program for a computer, or other computing device, that can emulate the BIOS and bootstrap loader of a video game console or handheld, so a computer can be used to play, hack, translate, or modify games that were created for that console or to develop games for that console. Console emulation can also be done between consoles (hence cross-console emulation), making a modern video game console emulate a less advanced one.
Commercial console games for emulators are often distributed as ROM images on the Internet. Without the permission of the copyright holder or the Entertainment Software Association, this practice is illegal, though few copyright holders have even released their games and demos gratis or even as free software. This illegality is also controversial for long time gamers and so called "old school" gamers. The popularity for console emulation among fans is due to the belief that many older video games that are no longer on the market are more enjoyable than newer video games currently on the market. Part of this comes from game companies having to focus on gameplay elements other than graphics, due to the graphical memory and hardware limitations at the time of the 8-bit and 16-bit eras, while some believe that modern 3D graphics have not yet fully matured. Another belief is that companies can no longer make income from older titles, though this is not always the case with published archived collections, ports and rereleases, emulated titles, and enhanced remakes provided by the original publisher or copyright holder. On some of these "Emu" sites, there is a myth that you may only keep a ROM image on your computer for a period of 24 hours. This idea stems from the belief that ownership of the actual game is not required.
Keeping a ROM for less than 24 hours is legal
Some ROM sites put a disclaimer that keeping a copy for 24 hours for "evaluation purposes" is legal; there is no actual legal grounding in this claim.
Having a physical backup makes ROMs legal
Many ROM sites claim that it is legal to download the ROMs for backup purposes if one owns a physical copy of the software. It appears that Title 17 USC Section 117  permits making a backup copy within the United States, but this has never been tested in a court of law. So far, this is still in the legal gray area as owning an Audio CD but downloading MP3 tracks ripped from it from the Internet.
Dumping your own ROMs is illegal
This point sort of continues where the previous one left off. Just as it is legal to take a CD you own and rip it to MP3 yourself, it is legal to take a cartridge you own and dump it yourself (in fact, this is the only legal way to acquire ROM images without express authorization from the copyright holder). Nintendo especially likes to further this myth.
Emulators themselves are illegal
Emulators are only illegal when one of the following situations occurs:
Outside of these circumstances, emulators are legal. Nintendo especially likes to further this myth as well.
Importance of Emulators
One advantage to ROM images is the potential for hacking: Console game fans sometimes produce translations of games, rewrite the game's dialogue, or apply fixes to bugs that were present in the original game. Software which emulates a console system can be improved with additional capabilities that the original system did not have, such as variable-width font, anti-aliasing, or game save state functionality.
Emulators such as Genecyst helped fuel the early days of the Sonic scene by including debugging tools along with regular game functionality. Allowing users to see palettes and currently loaded tiles, for instance, helped initially discover the "Press Start" message that was hidden within Sonic 1.
However, the most important contribution emulators made to the early days of Sonic research was their ability to create savestates. Asides from a few emulators such as KGen, Genecyst's savestate standard of *.GSX allowed for a simple format to start studying palette and item values. The knowledge learned from these early hacks allowed for the full-fledged ROM hacks which are prevalent in the scene today.