PNR stands for "passenger name record," and it's the Trenitalia booking code for your train ticket. You receive a PNR when you have purchased an e-ticket.
On some trains, the PNR is all you'll need to show to the conductor (either printed out or on your smartphone or tablet). On other trains, you'll be required to use the PNR in a self-service ticket machine at the station to print out a paper ticket before boarding the train.
Be aware that if you print a ticket in a rail vending machine you will have to validate the ticket prior to boarding the train See: What does it mean to validate a rail ticket or a rail pass? What you need to do will be indicated on your confirmation email.