Multiple players has much more to do with your internet connection than your computer. As long as your computer can run the game then you are golden on that front, and I think you can look that information up yourself.
Now, if you are playing a game on the internet, then you can typically do just fine so long as you have an uninterrupted 5mbps connection and a ping rate under 30ms. Having a little less bandwidth or a little higher ping rate will still work, but you may get a little noticeable lag on occasion. Having more bandwidth and lower ping is always suggested. But as the game is hosted somewhere else, it is no big deal because you are only downloading one set of information for all of the other character's movements.
Now if you are playing a game where you are hosting it, such as Minecraft, or many older FPS games, then you want a much faster connection. Something along the lines of 15-20mbps and a ping rate under 20ms (preferably under 10 if you can muster it). You need all of this bandwidth because you are going to receive and send information for each and every player connected to you, and any issues that you have will then cause issues for everyone connected to your computer.
Often times the ping rate is much more important than throughput. A cable or fiber connection is highly suggested if you are hosting. DSL is simply not going to muster a fast enough ping to keep everyone updated fast enough.