MAME, or the Multiple Arcade Machine Emulator, emulates the hardware that powered old coin-operated arcade games machines - Space Invaders, Pacman, thousands of the old favourites - which may allow you to play these on your own PC. It's a great way to bring back the memories of your youth, as they'll look and sound almost exactly the same.
There are complications, however.
The first is that to play these ancient games you'll need images of their ROMs, and perhaps other system files. While MAME itself is legal, distributing the ROMs usually isn't, as most are still the property of their creators. There are sites that make these available, but it can take a while to find the required files for a particular game.
And the second is that, although MAME is technically brilliant, it's not exactly designed for ease of use. It doesn't come with a graphical front end, for instance, so by default you must extract the programs files to a folder somewhere, read the TXT based instructions, download the other files you need, place them in the right folders, then use the command line to view and launch your games.
If you're an experienced PC user then it's worth doing this, at least initially, to give you an understanding of how MAME works.
Otherwise, though, you'll probably want to get some help on using MAME. The program's WIKI site has useful advice, and third-party sites like MAMEWorld can point you to ROMs, graphical front ends, and other tools that will make your game-playing life considerably easier.
See What's New for more information about this release.