ps3 fat was the original design ranging from 40-80 gb versions i believe some didnt have wi-fi...it had a shiny case and power/eject buttons that you touched and didnt have to press on...40-60gb versions were backwards compatible in the beginning
the ps3 slim new design is more standard...nothing really amazing about the flat black case..thinner for sure...larger hard drives, wi-fi on all of these...
basically you buy a "fat" and your buying an older version and im not even sure they still sell them in stores.....
i have a fat and a slim and they both work
just buy a slim and save yourself from anymore dumb questions