If you're asking about the movie preds, most are just named by their bios, though a few have their own names.
The first movie, most just call P1. The second movie, there's P2 as the main. Then there's also the Lost Hunters at the end of the movie. The P2 elder (AKA Ancient, Grayback), Gort, Snake, Borg, Boar, Brother Boar, Ram, and Hippie. I may be missing one, but I just got off work and I'm tired.
In AVP, there's Scar, Chopper, Celtic, and AVP Elder. Though, there is are a total of 43 other preds in AVP that it shows but are never named, and most aren't shown close enough to even make out a bio shape. Yes, I went through the movie and counted the preds in the flashbacks and at the end standing outside the ship. In AVP:R, the main pred is Wolf. I think in one of the scenes of the homeworld, it shows a few others, but I can't remember. Horrible movie. Only good thing about it was Wolf. And then there were several shown at the begining and in deleted scenes who died on the ship.
And then the newest batch from Predators... The three main ones are Berserker (Aka Mr Black), Falconeer (Mr. Plow), and Tracker (Tusky). The pred that looked like the P1 is called P1S by a few of us here, I don't know if he has another name. Then there are countless in the comics and books like Elkman said before. As far as I know, that's pretty much all the common names. Though I've heard people call P2 "Pussyface" and P1 called "Anytime."
Anyone feel free ta correct me if I'm wrong.