I'm assuming you're intending to play legit PS2 game copies. :0
http://pcsx2.net/
As for how well games will play on various computers, I dunno. I've got a 6 year old Core 2 Duo 2.4GHz CPU and a Radeon HD 5770 graphics card, and I was able to play my friend's copy of Ar Tonelico quite nicely when we tested it.. but then again, it's a 2D game, so it might be easier to emulate.
You can probably get by with a weaker graphics card than mine, but CPU-wise, you want at least a dual core of respectable power--mine apparently is good enough.