"Posi" is short for Positive Traction, which is the trade mark name for the Eaton clutch type limited slip differential. The name has been applied to all limited slip differentials over the years, much like Roller Blade has been applied to all brands of inline skates, and Kleenex to all brands of facial tissues. It's not correct, but that's just how it is. So the short answer is, Posi means limited slip differential.
Most LSDs require a special oil additive to help the clutches slip smoothly when turning corners. It reduces the clutch chatter and makes them operate more smoothly.