It's awkward, because there are so many things that Pyramid head COULD be, that I don't think we'll ever know which one is right.

It could be, as has been said, a representation of James himself, or a representation of his guilt and his wish to be punished. It could be a projection of Samael and/or (SPOILER)Walter Sullivan, or could be the Priest of Valtiel that Seifer suggested.

BUT, I'd like to point out this :

Pyramid Head doesn't just punish and try to execute James, think about the first scene where you see him, he's 'raping' those marionettes in the apartment room, and wherever you go he kills everything in his path, not just tries to kill James.

NOW, that could mean even more evidence that he's a representation of James, if you think about it, because James' sin was his abhorrence of ugliness, and since all the enemies are considered ugly then Pyramid Head could be doing what James would do, perhaps.

On a slightly off topic thing, I have to ask...

Spear Pyramid Head vs Great Knife Pyramid Head?