An emulator is hardware or software that enables one computer system (called the host) to behave like another computer system (called the guest).
For instance, Sega Mega Drive and Genesis Classics running the Sega Genesis title Sonic the Hedgehog (1991) on any current 20xx era PC.
- Emulation General Wiki - A large wiki dedicated to video game emulation
- Video game console emulator at Wikipedia
- List of video game emulators at Wikipedia
To run software using an emulator, one usually needs the following:
- Emulator program binaries for the host platform
- Original software, copies of which are sometimes called "dumps"
- BIOSes or firmware copies, when applicable
Emulation is an extremely cost-effective way to distribute old games for more recent hardware, so it increases chances of old games being officially re-released. Re-releases / remakes using emulators are gaining ground and visibility on official stores.
- Gametap (now defuncted) offered emulated games.
- On consoles, Nintendo's Virtual Console and Sony's PS2 Classics are emulation-based services
- DOSBox is used by Steam and GOG.com for almost all DOS games
- Neo Geo games available on Humble Bundle are pre-configured emulator+game packages.
Emulation-based official releases almost always fail to provide the following information, that PCGamingWiki game page may document:
- compatibility with any other emulator 
- emulator release used, hardware configuration for PC games
- exact version of the game: PAL/NTSC release, World/USA/Japan/Europe/other release, disk/CD version, etc.
Emulation allows modifying or extending the behavior of the original software, for instance:
- Resolution increase and custom textures for 3D hardware-rendered games
- Netplay (for games that originally offered couch co-op)
- Game modification and translation, via unofficial patches
- Input-based gameplay record and playback. Emulators can be very convenient for speedruns and TAS in particular
- Hardware customization. DOS games on GOG.com or Steam are pre-configured, but one is free to change the guest sound card or the display device, since these are virtual and handled by the emulator. This is explained in greater detail in the DOSBox article.
Official re-release providing dumps of the original software in clear (unencrypted / unobfuscated) means that the game is in effect DRM-free and not locked to the bundled emulator so that it instead could be used on another emulator for the host platform (one which provides netplay features for instance), or any other host platform given there exists an emulator for the guest system on it.
Last, unobfuscated DRM-free emulation-based releases are the most convenient way to acquire backups for the games.
An alternative is to buy the original game, and dump it using an appropriate device. However, it could be less than obvious depending on the hardware/game, and making private backups is not something that every jurisdiction in the world allows.
Goals and legality
Backups were mentionned above: emulation is important for preservation, as game companies often fail to properly preserve the games and document them for later generations.
Emulators are considered legal to develop and to distribute, as long as they are not bundled with copyrighted software. There are a few precedents, such as Bleem! vs Sony case for example.
One can try to check the game installation directory for an executable file that looks like an emulator name or short name. If you find such a file, you may be able to then identify the emulator version by checking the binaries properties, such as the Product Description on Windows (right click > properties). Most of the time, the packaged emulator will be a custom one, though, so it is not obvious to know what it actually is and how accurate it is.
Some emulators have a built-in database of game dumps they know of, which allows precise identification of a game or software by automatically comparing the entries in the database against a file provided by the user.
MAME is one of the emulators with such functionnality.
Here is an example with Altered Beast from SEGA Mega Drive & Genesis Classics (dumps are in the "uncompressed ROMs" installation subfolder):
mame64.exe -romident ALTEREDB_UE.68K Identifying ALTEREDB_UE.68K.... ALTEREDB_UE.68K = mpr-12538f.ic1 megadriv:altbeast Altered Beast (Euro, USA)
This tells us that this is compatible with MAME and that the game dump corresponds to the Europe (Megadrive) and USA (Genesis) release of Altered Beast (and we don't have the japanese release available). "megadriv" is the name of the emulation driver, "altbeast" the short name defined in MAME for the entry.
Note that MAME database does not keep track of bad dumps, which are discarded when found. If a bad dump was found, "NO MATCH" would have been the output. Bad dumps are bad copies or hacks, however, MAME and other emulators may be able to run them.
List of PCGW emulator pages