I believe you're asking about backwards compatibility. While I agree a PS3 should be able to play PS2 games, it's sadly not always the case. While they all can play PS1 games, not every PS3 can play PS2 games. PS2 compatibility depends on the model.
Backwards compatibility with PS2 games has slowly been phased out. The original PS3's had hardware emulation (quite literally there is a PS2 inside of your PS3). The later revisions had software emulation only (PS3 processor pretends to be the PS2 processor), and then none at all. No slim PS3 has backwards compatibility.
20 GB PS3 Fat/Original - Model CECHB01
60 GB PS3 Fat/Original - Model CECHA01
80 GB PS3 Fat/Original - Model CECHE01
In my personal experience, there's no real difference between software and hardware emulation in terms of actual game play. Software emulation has a few hang-ups on some older games, but overall it works pretty well. And it's infinitely better than not having any ability to play PS2 games at all.
Links!WikipediaHardcoreware.net - Comparison TableSony - check compatibility of games with PS3
Hope that helps!