Your Model
Only three PS3 models feature PS1/2 compatibility. These are the original 20GB model, the later 60GB model and a single variant of the 80GB model. To check the model of your PS3, look for a printed serial number on the back of the machine, to the left. The seven characters after the hyphen are what you're looking for--the backward-compatible models are CECHB01, CECHA01 and CECHE01.
Your Game
Not all older PlayStation games are playable on backward-compatible models of the PS3. Sony's website provides a utility for checking the compatibility of a particular game (see Resources). At the site, select your PS3 model under "Platform," and enter a game title to check its compatibility.
Suggestions
If you do not have a backward-compatible PS3--or if the games you want to play are problematic on the PS3 or simply don't run at all--your best bet might be to pick up a cheap used PlayStation 2 online or at a game store. PS2s are much more reliably backward-compatible with PS1 games and are now old enough that they are not prohibitively expensive.