Different models of the Sony PS3 support different levels of backwards compatability with Sony PS2 games.
Sony PS3 - 20GB and 60GB NTSC Model
The 20 gig and 60 gig NTSC models of the Sony PS3 are the absolute best models for backwards compatibility with the Sony PS2.
These models contain PS2 circuitry in hardware. Each of these units contains a full PS2 Emotion Engine (EE).
Unfortunately, this hardware-based approach was very expensive and Sony discontinued the 60 gig PS3 after a very short time.
Sony PS3 - 40 Gig Model
The 40 gig model of the Sony PS3 does not support any backwards compatibility with the PS2.
However, the 40 gig model still maintains full backwards compatibility with the Sony PS1.
Sony PS3 - 80 Gig Model and 60GB PAL Model
The 80 gig Sony PS3 supports PS2 games through software emulation.
The success of this emulation was poor at first, but Sony has improved it with several generations of firmware upgrades.
If you are experiencing difficulties with PS2 emulation, it is recommended that you upgrade to the latest firmware.
The 60GB PAL model also utilizes software emulation.
Check PS2 Game Compatability
Sony provides a Playstation 3 Backwards Compatibility Status database which enables you to find the current status of backwards compatibility for specific video games:
http://www.us.playstation.com/Support/CompatibleStatus