MAME stands for Multiple Arcade Machine Emulator. When used in conjunction with an arcade game's data files (ROMs, CHDs, samples, etc.), MAME attempts to reproduce that game as faithfully as possible on a more modern general-purpose system. MAME can currently emulate many thousands of classic arcade video games from the the very earliest CPU-based systems to much more modern 3D platforms.
The main purpose of MAME in short is to be a reference to the inner workings of the emulated arcade machines. This is done both for educational purposes and for preservation purposes, in order to prevent many historical games from disappearing forever once the hardware they run on stops working !