Short for 'piss' when used in the sentence 'get on the piss' (get drunk). The amusing part of this abbreviation is that P is also a drug (aka Ice, methamphetamine, crystal meth) with a very bad image portrayed by the media. So when in public and you ask your friend if they want to get on the P tonight, you generally get a funny reaction from people around you.
Caleb: "Hey bro, wanna hit the P tonight?"
Alex "Yeah man its been like 4 days!"
Old people "Oh my god, druggies! Call the police!"