Who is the mysterious G-man in the Half-Life series?

There is not a clear explanation of this. There's not really even a good solid HINT of this. This is entirely deliberate on the part of the game developers - they haven't revealed SQUAT about this guy.

The Nihilanth was of the opinion he wasn't human, though: "You are man. He is not man."

There's a pretty good summary of everything we have at the Half-Life Wikia but I doubt you'll learn anything you didn't from playing the games (except possibly locations he shows up that you missed him).

In HL2: EP2, after you bring back the antlion larvae to heal Alyx, GMan starts talking with you for the last time in the series, mentioning something like "he wished he could do more than keep an eye on Gordon and Alyx, but he needed to ... FOLLOW SOME CERTAIN RULES."

This statement still bugs me since the first day I heard it, because I always thought on GMan as a "superior entity" or something, but that statement may mean he's following orders of someone... or something else. Could he be an advisor's spy? After all, he KNEW that Eli would be attacked by advisors since he whispered into Alyx's mind to tell her dad to "prepare for unforeseen consequences". And Eli DID meet GMan before.

Also, there's still another important fact. There's still a theory that when Gordon pushed that cart away in black mesa, nothing of what we know happened in real life, but inside his mind, he could be suffering of a sickness where you "dream awake".

And BEFORE we get to that chamber, all that Gordon sees are guards and scientists... but a little before reaching the chamber, he sees gman talking to a scientist. Meaning he was the ONLY DIFFERENT THING in entire black mesa, and maybe his image buried so deep into Gordon's mind that when he fell into sickness... he sometimes sees GMan in the background, or even talking to him.

Well, there's really no proof of any of these theories, but Half life canon is always fun to argue about.