In the RE2 rules for the Playstation version, it says, "Show Playstation Driver after a run is completed". What exactly is a Playstation driver and how can I "show" it?

You can show the driver on the system configuration menu, you get to the screen when a disc isn't inside the console.

So after you complete a run open disc tray/restart console

It should be noted this is only for when playing the PS version on a PS2, and a PS2 with Fast Disc Speed option at that.

Oh I see, now I remembered where it is, thank you very much. I plan to do a run on a PS3 most probably.

Good luck friend!