A PS3 can play at least every Final Fantasy from 4 to 13-3 except for XI and XII. An original PS3 (with PS2 backwards compatibility which was discontinued) can play XII and technically XI as well. I am not sure about XI considering the MMO elements.

With backwards compatibility you can play:

Final Fantasy Chronicles (Contains FFIV and Chrono Trigger)  
Final Fantasy Anthology - (Contains FFV and FFVI)  
Final Fantasy VII through IX

PS3 versions include X, X-2, XIII, XIII-2, and XIII Lighting Returns