The Peterson-Lip (P-Lip) refers to the lip part of the mouthpiece. The hole of the P-Lip points upwards of the bit, towards the smoker's palate as opposed to classic fishtail bits pointing towards the smoker's tongue.
Peterson introduced this special bit in 1898. ▼

▼ Savinelli's (Italy) "Dry System" line is fitted with a P-Lip mouthpiece. The hole has the shape of a rectangle, this time.

▼ Mauro Armellini's P-Lip (Italy) has a smoke hole which seems to be drilled straight into the bit. These mouthpieces used to equip the "Mauro" pipes, a second of Armellini. This pipe looks very close to Peterson's shapes.

▼ Some of the pipes in Vauen's Dr Perl line (Germany) are equiped with a conventional P-lip stem.

▼ The drilling of this Bontemps pipe bit obviously isn't performed according to the P-lip principle. The smoke isn't deflected towards the palate. This stem probably was crafted by the "L'Ebonite" factory (Paul Jeantet & Cie) in Saint Claude France.

▼ The mouthpieces of the WDC's "Wellington" line culminate in a P-Lip very similar to Peterson's.

▼ An original mouthpiece of the almost unknown "Swiss" brand.

▼ The Chesterfield either from Kaywoodie or from Yello-Bole is a knockoff of the Peterson P-lip pipe.

What they say about P-Lips in the forums

Ofafeather: "My dad never liked them but he said it's probably because that have a sort of phallic look."

Mongo: "When I clenched a pipe with a p-lip, the hole in the p-lip always ended up right against one of my teeth. The hole was completely blocked by a tooth. This happened, amazingly enough, on both sides of my mouth.
I could clench the pipe without blocking the hole if I moved it deeper into my mouth, but I found that to be a singularly uncomfortable way to hold a pipe. I finally gave up, and drilled a hole straight into the bit."

Thefes: "My problem with the P-lip is that it seems harder to get a pipe cleaner through it. It seems that the wire tip of the cleaner wants to dig into the wall rather than slide through nicely."

mpm. Bandit (transl. from French): "I 'm a clencher and, honestly, p-lips are not an issue..."