Other people have already discussed how you can use MAME to play all your favorite games on your personal computer. So I won't bother going into that. Instead I am going to talk about the true purpose behind MAME.
The real goal of the MAME project is to preserve old arcade games for future generations. There have been around 7000 different arcade games released since Computer Space started the whole thing back in 1971. Many of those games are still fairly common today, while others have already become rare, and hundreds of titles have literally become extinct. You see most arcade games were designed to give about 3 to 5 years of service, before being replaced. With repairs they can last much longer, but they do not last forever. Every year literally thousands of games (and game circuit boards), are destroyed, parted out, or become unrepairable, In fact some games even have a suicide chip that will render the games PCB (printed circuit board), unusable after a period of time. Capcom titles are especially bad in that respect. Eventually every single Final Fight and Street Fighter II machine will stop working because of this (those are just two titles I picked randomly, there are hundreds of others). They are like ticking time bombs, eventually they will go off.
Suicide chips are not the only problem facing arcade games. Bit rot, heat damage, or a simple power surge can render a game inoperable. These faulty boards can sometimes be repaired, but in many case they cannot be. They are often simply discarded because replacement game boards will often be cheaper than the repair would be. Hundreds (perhaps thousands), of games die every day all around the world. These dead games cannot be replaced (except for a few very popular titles that have reproduction parts available for every single component of the machine, but those games only represent a small fraction of the titles that are out there).
Over time the "population" of a game will constantly, and steadily decline (except for a few cases where people are making more of them, see the Ms. Pac-Man node for details). This is already apparent with many games from the early to late 1970s. There simply are not any working copies left. Arcade games simply cannot be readily copied to keep them alive. Music, movies, and books all survive the ravages of time because they can be copied. But you can't just stick that Galaga game into a VCR and make a copy of it. It just doesn't work that way.
Lets talk about a specific title for a minute. Back in the early 70s Atari made a game called SHARK Jaws (under their pseudonym "Horror Games"). This game was historical for several reasons. The first was that it was one of the first games to be based on a movie. It was based on Jaws (although it was an unlicensed translation). The second reason was that it was one of the first (perhaps the first) games to use animated bitmap graphics. There were several thousand copies of this game made, so you would think that there would still be a lot of them around. But there are not. Do you know how copies of SHARK Jaws are known to exist? I will tell you. There are only two known copies left of this game, and only one of them works. The first copy is owned by the Videotopia arcade exhibit (this is the one that works). While I own the other one, which is complete except for the cabinet itself (but the cabinet can be rebuilt, as Atari provided full cabinet specs in the manual, and I have literally every component, and piece that bolts onto the cabinet). I may never get mine working, and Videotopia's copy could stop working at any time. When that day comes, another historic game will have been lost to the ravages of time. The only hope for saving this game is the MAME emulator. (It is currently unemulated, but I will gladly send my circuit board to the MAME team if they need it to get this game working). Without MAME, this game will die. And it is only one of thousands of games that face certain death without MAME.
Right now MAME is looked down upon by most arcade game collectors. They think it cheapens the hobby. But, many of them don't realize that MAME is going to be the only thing that can truly allow their beloved games to last forever. How many working Ms. Pac-Man machines will there be in another 20 years (remember, they are all already 20 years old), how about 20 years after that, what about 50 years after that. A properly cared for game cabinet will last forever, and the industry standard joysticks, monitors, and power supplies are still being produced, and probably will be for a long time. But the game boards are not being produced anymore, and eventually they will all die. But MAME can save all those machines from an untimely death. It is quite simple to swap a MAME computer in place of a missing or defective game board, restoring new life to a machine that would otherwise never eat another quarter again. Eventually MAME will be the only repair option for many machines, and luckily it is quite simple to implement. This is the true future of MAME, people may always use it to play old (and not so old) games on their computer. But as time wears on MAME will prove invaluable at replacing all those game components that are simply irreplaceable by any other means. I would go so far as to say that fifty years from now, most of those 1970s to 1990s arcade games will only truly exist in MAME format, as all the original boards will have stopped working decades before.
I will close with an example of a machine that I own, that may some day be saved by MAME. I have a dedicated Special Criminal Investigation upright. This is a driving game from the late 80s. There were several thousand copies of this game made, although most were sold as conversion kits. There were four different dedicated cabinets made for this game, all of which are uncommon individually. This title has not one but two suicide chips, and it is already difficult to locate working boards for it on the aftermarket. My copy of this game is in mint condition, but is missing the circuit board that actually contains the game. Even if I could find a board, it would die eventually due to the suicide chip, or a decade or so later due to bit rot. MAME will be able to save this game. With a MAME computer in the place of the missing board, this game will be playable again. Once again people will be able to drop a quarter in, hop in the squad car, and chase off after the criminals. Without MAME, it would never be possible.