M.A.M.E. (Multiple Arcade Machine Emulator), or short MAME, is a software emulator, allowing to play original video games from the arcades, starting from 1975 to today and tomorrow. Its main purpose though is to preserve decades of software history. Besides the emulator itself you also need roms from the game you want to play.
An emulator in general is creating a hardware environment (called "driver" in MAME), like chips in a video game (CPU, RAM, I/O and other support chips) by software. Thus you can "have" what ever video game hardware you want in your PC or smartphone. Then you can load up the original code, usually dumped from the original chips of a video game, into the emulator, and play them. Since it's all original you can even experience bugs and glitches which came with the original game. But you also have the option to activate cheats, speed up or slow down the game and more.
Since 2015 MAME's sister project M.E.S.S. (Multi Emulator Super System), or short MESS, was merged into MAME. MESS allows to also emulate any computer, if somebody wrote a driver for it and the BIOS rom was dumped. You could run an Atari 2600, Commodore 64, Amiga, Atari ST, (Super) Nintendo Entertainment System, SONY Playstation, XBOX, SEGA... Well anything existing. In theory you could play on a Playstation or other which would only exist in say three years - if someone would get hold of it specification. But that's just theoretically. And even if, no actual of today's PC hardware is powerful enough to run it in a decent speed. Even on older systems or video games modern PCs cannot run the emulation in a speed to match the original speed of the emulated machine. There are also prototypes or hard to come by systems you can emulate. Like the Apple I or Commodore 65 you can run.
If you want to bring back some fond memories you had when you had your first game console, first home computer or video games you played as kid in the arcade back in the day, the MAME emulator brings back nostalgia.
Legal mention: Obey the ROMs copyright!