Only one type is backward compatible with some PlayStation 3, and that is a certain type of 80gb that is no longer in production.:{:{:{:{:{:{:{:{
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First Generation models ie. those that were released first were fully backwards compatible and played PS2 and PS1 games as well as PS3 games. The second generation of the PS3 also which also had 4 USB ports were also Backwards compatible with many of the PS2 games. All other PS3 models lack the backwards compatible feature and are not considered to be Backwards compatible. see related link.
yes No you can not turn a PS3 into a Backwards Compatible PS2 game player. That's why the demand for older Backwards Compatible PS3 models with the 4 USB ports has increased the resale value. If you could make a Backwards Compatible PS3 Slim 250 GB why would someone pay as much for an old PS3 with an unknown history that might have been repaired many times.
If you PS3 is not backwards compatible nothing you can do will change that
Hate to tell you this, but every ps3 can play ps1 games including the slim. 20gb 60gb 80gb are backwards compatible with ps2 games.