Only specific PS3 models can play PS2 discs natively. Here's the complete breakdown.
All later models (40 GB, 80 GB non-MGS, 120 GB, 160 GB, 250 GB, 320 GB, Slim, Super Slim) cannot play PS2 discs.
Look at the barcode sticker on the back: "CECH-XXXX". First two letters after CECH indicate the series.
If you have a launch 60 GB or 20 GB "fat" PS3, you're set. Otherwise, use one of the alternatives above.