I'm certainly no X's and O's guru, but the brief 411 is that the Princeton Offense was the offense that Pete Carril utilized at Princeton for many years. It relies on cuts, not screens and usually results in scoring off of backdoor cuts (for layups) or three pointers. The center sets up shop in the high-post rather than the low post, which opens up the lane for players to make cuts to the basket.
That's a very high-level explanation.
I'll never forget the game where Princeton almsot beat Georgetown in the first round of the tournament when Georgetown had Mourning and (who-wants-to-sex) Mutombo. It really was a thing of beauty. They would be passing the ball back and forth along the perimeter and to the high post and then all of a sudden some seemingly unathletic white guy would get a pass off a backdoor cut and an uncontested layup. It kinda seemed like magic.
My beef with the Princeton Offense is that I think it is a dumb offense for teams with good athletes. It is a slowly developing, deliberate offense. The result is a game with fewer possessions, and a game with fewer possessions is a game that can more easily be won by the inferior team. It keeps games close, which is a silly thing to do when you have athletes to win by 30.
Reasonable minds could differ on that, though. Just my take.