Whatever the game you are playing, the High Concept is a core aspect of any character. It speaks to who the character is deep inside. In essence, it can be seen as something of the "trope" that the Player is looking to fit into, while the other aspects or things done with a character refine that trope into a truly original character.
As I work towards making a hack for FATE: Superheroes (not that there aren't already tons out there), I find myself trying to go through and code a character up as I play around with the rules. The fun part is, this character to me has effectively become a concept themself in my head.
The God Damn Batman
Batman may actually be the best example of this. Batman is an idea, a concept, and the character personifies that. If I put "Batman" as my high concept on any character sheet the GM would know what I was going for, to a certain extent, because of that very thing. However, if I were playing Batman - and he weren't as famous as he was - I wouldn't be able to put down Batman as the concept now could I? You can't have your character's concept be there own name. It doesn't tell anything about the character - well, aside from their name anyhow. So what do you do?
Boil It Down To The Basics
For this you need to boil the character down to the basics. As small as you can get them, what is at the core that you want to convey with the character? You can see this with all the fan renditions of Batman himself. Is he a "Billionaire Playboy turned Vigilante," "The World's Greatest Detective in Cape and Cowl," or "Emotionally Scarred Control Freak Trying to Protect His City?" All three of those are perfectly valid for Batman as a character, and all three speak to different core aspects of the character that we could use to explore. Amusingly, none of those are the ones I think I'd use for Batman if I made him. Instead I'd use "Highly Trained and Motivated Protector and Avenger of the Night."
Still, look at the four of those. Like I said, they all point to different takes on Batman, Bruce Wayne, and the fusion of those two identities and that is ok. Every writer writes Batman a little different based on their own view of the character and what makes him awesome. Why should you be any different? Heck, just in the movies we have everything from the zany/light hearted Adam West Batman, the completely insane Michael Keaton one, and then the Val Kilmer, George Clooney, and Christian Bale takes on the same character. None is more or less authentic or real than the others, it just matters which fits the story you want to do.
Refine With Other Aspects
Once you have the core of your take on the character, you can refine the character further with the other aspects. Maybe you want to point out that you're ridiculously wealthy for your Batman character and so you throw on "Last member of the richest family in Gotham" or "The Prince of Gotham" onto the sheet. Maybe you want to play up the detective training and use "World's Greatest Detective." Maybe you want to play up the complex angles for social relationships with something like "Father, Son, Teacher, Asshole."
The point is the other aspects, or other defining traits if not using Fate, are what throw the meat onto the bones and make the character unique.
The Final Result
So what about me and the character I'm working on? Well, for the High Concept to boil it down to the basics I need to go back to City of Villains where the character started off. In that game the character was defined as being a "Super Strength/Invulnerability Brute." Coding that for RPGs I'd probably change it to "Super Strong, Invulnerable Brute" and go from there. The two power designations in the concept give the core of what the character can do, the Brute gives a good grasp (even if they're not a stereotypical brute) on how they approach problems. I can then use other aspects to help refine the character.
In the end, their aspects could easily look like:
High Concept: Super Strong, Invulnerable Brute
Trouble: I'm Always Up For A Fight
Aspect 1: Definitely A Top Tier Threat
Aspect 2: Not As Dumb As I Look
Aspect 3: I Never Said I Was A Villain
All in all, not too bad for the character, if I do say so myself.