MAME stands for Multiple Arcade Machine Emulator. When used in conjunction with images of the original arcade game's ROM and disk data, MAME attempts to reproduce that game as faithfully as possible on a more modern general-purpose computer. MAME can currently emulate several thousand different classic arcade video games from the late 1970s through the modern era.
MAME is free. Its source code is free. Selling either is not allowed.
MAME is strictly a non-profit project. Its main purpose 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. Of course, in order to preserve the games and demonstrate that the emulated behavior matches the original, you must also be able to actually play the games. This is considered a nice side effect, and is not MAME's primary focus.
If you would like to see a more complete list of changes, check the whatsnew.txt file located here.