If you just want classics (any games from 70s and early 80s) you can get a Pentium 3 500MHz and 128 or 256 Megs RAM and the games will work fine.
If you want some Street Fighter/Mortal Kombat and later action in addition to that, Get a Pentium 4 or AMD equivalent and at least 256M RAM.
My personal MAME cabinet is running an AMD Sempron 3000 Processor (2GHz) (Socket A) and an ECS motherboard that came with it as a combo deal. It has 512 meg stick of RAM and it will run all the games that can be currently run on MAME. I do mostly classics, but I do try newer games that I never played before.
SOME of the newer games (1995 and later) may not run 100% even on a system 2 or 3 times as powerful. It is just the nature of the emulation. These games need something like a 10GHz processor or something ridiculous like that to run properly in MAME. Currently, nobody is really able to play these games 100% on MAME, so with those games, we will just have to wait for the hardware to catch up.
Basically, if you go to the computer store and get the cheapest NEW computer that they have, it will be have than enough horsepower to run any of the games that are currently playable in MAME.
Or you can go to Newegg, or Tiger Direct, or Fry's or wherever, and get the cheapest CPU/Motherboard combo, use 512 Megs of RAM and put a basic system together for yourself, and that will be plenty for anything current in MAME.
Getting more expensive hardware will just be overkill unless you want to play Doom3, Halo, or WOW on it.
As for OS, just use regular old Windows XP. It works fine.