HUGE SPOILERS AHEAD CONCERNING THE PLOT OF SILENT HILL 2. YOU HAVE BEEN WARNED.
What is the Pyramid Head?
The pyramid head is the freakish boss character from the survival horror game Silent Hill 2. In a similar way to the Nemesis in Resident Evil 3, he (it?) follows you around the town attempting to kill you at various parts of the game. Most of the time he wields a large Knife called The Great Knife, which slows him down to a crawl as he has to drag it along with him (because of its enormity). Sometimes he has a spear, and with this he can run after you. His name comes from the fact that he has a rather large Pyramid shaped head covered in dry blood, with an apron also covered in blood.
Where Will I Meet Him?
The first place you will meet him is in the Apartment Building near the beginning of the game. You will see him brutally murdering a couple of monsters, and later on you will have to fight him on a stairwell. The next place you meet him will be at Brookhaven Hospital, where, in the long basement corridor, he chases you and Maria to the elevator with a spear. You later encounter him in the Toluca Prison and Labyrinth sections of the game, where he will be just walking around like any other monster (although you can't see him in the prison, you can most certainly hear him on the second floor, scraping his knife behind him).
The final place you encounter him is at the Lakeside Hotel, where you will have to fight two of them. After this they impale each other and die.
Okay okay, but what exactly IS he?
Tough question. The first clue as to his origins is the picture in the Silent Hill Historical Society building, right before entering Toluca Prison. On one of the walls in the first room, there is a picture of the Pyramid Head with lots of hanging corpses in the background. Underneath the picture, the words "Misty Day, Remains of the Judgment" are printed. It is quite obvious from this picture that the Pyramid Head must be some sort of executioner figure, and is connected with the town's dark history.
Once inside the Prison, you discover more hanging corpses, and ones of special note are the ones in the yard, where two Pyramid Heads stand either side of two hanging corpses. At the end of the game, Maria is executed by the two Pyramid Heads you must fight in a similar way (as well as a few other times before), who then later kill each other.
From the information gained here and in various places inside the prison, we can safely assume that the Pyramid Heads are a creation of the town Silent Hill, based on the shameless executioners of olden days. Remember that Silent Hill is a place where your guilty thoughts come back to haunt you, and the Pyramid Heads are a remainder to James that he is a murderer, no matter how he chooses to believe it. The way the Pyramid Heads continually kill Maria throughout the game is a constant re-enactment of James' mercy killing of his wife. Now you know why Maria looks so much like Mary.
Okay, How do I kill the Bastard?
You can't. As a matter of fact, I believe that you never actually defeat him literally throughout the game. He either walks off or, in the case of the last encounter, kills himself. So he are some strategies:
Apartment Building Encounter
Your best bet here is to stay on your toes. Run to a corner and start blasting him with all you've got. Once he gets too close, head for the opposite corner. Use your speed as an advantage, as him dragging that great knife slows him down a lot. Don't get to close though, or he'll do some major damage to you. Continue to do this and a siren will be heard in the distance, and he will walk away. Don't follow him, or you'll die.
Another strategy, although I've never tried it, is to simply avoid him for as long as possible. Apparently the siren is sounded if you shoot him or not. But I have been told that this method takes longer, so it's up to you how you tackle it.
Brookhaven Hospital Encounter (basement)
As you descend into the basement, you will be in a large grey corridor with Maria. When you walk forward a bit, Pyramid Head will appear behind you (the biggest shock of the game for me) and Maria will tell you to run. You can't do anything here except run. Maria is in your way so you can't hurt him, and if you stop you will both get impaled. Once you reach the elevator, you're safe again. Except for Maria of course, who gets impaled either way.
Toluca Prison Encounter
If you listen closely while you're walking by the cells, you can hear Pyramid Head walking about on the second floor above the cells. You can't shoot him, and he can't reach you, so don't bother trying to kill him.
Labyrinth Encounter
When in the Labyrinth (the area where the map draws itself), you may encounter Pyramid Head once or twice amongst the corridors. There is no warning as to his presence, so be careful. You can try to kill him if you want, but I made the wise choice and simply ran, not looking back. I don't know if you can defeat him nor not.
Lakeside Hotel Encounter
Here you have to fight two of them, so you must be extra careful here. Use the same tactic as in the first encounter, except that you MUST shoot them as there is no siren. Due to your improved weaponry, use the shotgun to take them down. The hunting rifle is too slow and you can't run with it out, so don't bother. Just keep running from corner to corner and taking a shot or two with the shotgun. Don't bother with your Great Knife, it's just way too slow (no matter how powerful it is). They will eventually walk into the middle of the room and kill each other.