The PS3 can natively play PS1 games off the original discs. You can even play them with HDMI output.
All models of the PS3 can do this, including the original PS3 Slim.
To play PS1 backups you'll need 3.55 custom firmware. Even then, you can only play PS1 backups off disc, UNLESS you ALSO have the Cobra USB dongle, which allows you to play them off HDD (either external USB HDD, or internal HDD).