Only certain PS3 models (the 20, 40 and some 80GB PS3s) had the PS2 backwards compatibility to play PS2 game discs. They have been discontinued and now they only sell refurbished models of these for about $200. Honestly it isn't worth it because the model is what's known as a "Fat." They are more likely to break down compared to the new "Slim" models which lack the PS2 backwards compatibility. But honestly, the PS2 still played the PS2 games way better than the PS3's that were backwards compatible simply because the PS2 was designed to play PS2 games. Not to mention PS3's are designed to play Blu ray discs (all games come on Blu ray, which can support 1080p graphics if the game is programmed to do so.) PS2 games aren't made on blu ray discs. Playing disc types other than blu ray in the PS3 isn't actually good for it because it's not designed for those game disc types. They make the system work a lot harder than it has to by forcing the system to spin the disc faster and making the laser move faster which can result in overheating or damage to the inside of the system.
If your gonna get a PS3, then please do it right and buy brand new. The models that could play PS2 game discs are no longer in production so the only way you can get one is used. Buying used is a very bad idea. The Slims are a thousand times better as well, for several reasons which I will explain now:
*NOTE: the Yellow Light of Death (YLOD) is a MAJOR hardware failure that can occur from overheating, HDD issues, CPU/GPU issues and other main component issues!*
1. The Slims are 33% smaller, lighter, quieter, faster and more efficient (efficient as in your'll save money on power bills compared to the Fat.)
2. Overall performance is far superior.
3. It is far less likely to overheat or get the *WHICH ALL FAT MODELS ARE OVERDUE FOR NOW AND ONCE THE SYSTEM GETS THE YLOD YOU CANNOT RETRIEVE OR RESTORE ANYTHING YOU HAD ON IT*
4. They have remade several PS2 favourites just for the PS3. They are available on both the PS Store AND on DISC versions (discs are Blu ray so it's designed for the PS3, They reformatted the games especially for the PS3. They will also say "PS3" on the game case. Both versions have better graphics up to HD 720p, support 3D (great for the Sly games), and support trophies. They're called "Collection." (examples include but are not limited to: Jak and Daxter Collection, The Sly Collection, Metal Gear Solid HD Collection, God of War HD Collection.) So seeing "Collection" you probably guessed that there's more than one game on the download or disc. Each one comes with the series for the most part (so at least 2-4 games on each download/disc.) Good deal. I myself have the Jak and Sly Collections on disc and its WAY better than it ever was on the PS2 because of all the things they did. They literally remastered every game.
5. If I wanna play a PS2 game I will use my PS2, it's not hard. And the PS2 plays PS2 games better than the PS3 ever did anyway because it's designed to play them. The PS3 had to have PS2 parts mixed in with PS3 parts which results in issues with PS2 games. PS3 games are designed for PS3 and PS2 games are designed for PS2. Its a very simple and awesome concept that works.
6. The Fat models are way more likely to get the YLOD. The Slim had a change made to the GPU. The original Fat models had their GPU running at 65nm. The first Slim had it reduced to 45nm, and the latest Slim model has it even less at 40nm. What does this mean? It means your PS3 is using way less power and won't die as easily or get the YLOD as easily.
7. PS2s use standard DVD ROMs as their game disc of choice. The PS3 uses Blu Ray discs. The PS3 was DESIGNED to play Blu Ray, NOT DVD/CD ROMs. Playing those disc types on the PS3 is actually quite bad for it. It forces the system to work a lot harder than it has to (it spins the disc way faster and makes the laser move faster) with Blu Ray it won't even break a sweat so to speak as its designed for it. This is actually a HUGE reason for PS3's (even Slims) to overheat (which my own Slim has almost done on me because of a DVD ROM.) You would never expect it but it's true, so avoid playing anything but Blu Ray on ANY PS3 system. Not to mention that PS2 game discs are easy as hell to scratch/ruin compared to Blu Ray discs. They are very hard to scratch/ruin compared to any other disc.
I think its painfully obvious why the Slim is way better than the Fat was. I don't need 4 USB ports either when I have chargers that plug in the walls to charge whatever I need to charge that's an accessory to my system. And the PS3 doesn't use as much power as the Fat despite being more powerful than the Fat, so if I wanted to I could easily buy a USB splitter for it. The Fat in its own right is great, but I'd pick a Slim any day PS2 backwards compatibility or not. That to me is useless when I (like most people) have a PS2, even if the graphics aren't as good.
Hope that helps.