First of all let me note that In this article I will be focusing on the darker renditions of Batman, not the candy-coated and plastic-nippled Batman.
What is it about Batman that makes him one of our favorite superheroes?
To understand the essence of the Batman mythos, we need to go back to the beginning and understand Batman's origins. As a child, he wasn't that special, just another spoiled rich boy. And the single event of his parents' murder changed all that. Not only did he now have a reason to take vengeance on the criminals -- true vengeance, not goody-goodness -- but he also had a reason to be truly afraid of the world.
I have always liked the fact that Bruce Wayne chose the bat as his symbol: the thing that he feared the most. If you think about it, the symbolism is beautiful. Bruce is becoming what he's afraid of: an angry vigilante. Only his self control keeps him from becoming what he hates more than anything: a criminal.
As I see it Batman is half-superhero, half-villain.
Let's turn to The Christopher Nolan imagining of Batman and the reason why Christian Bale works so well as Batman in the Dark Knight trilogy is because Bale, quite frankly, looks like an angry guy. Bruce Wayne/Batman is an angry guy, full of repressed vengeance and bitterness. You can see all of this clearly in Batman Begins when Bruce Wayne brings a gun to Chill's parole hearing. He almost becomes a criminal himself, the thing that he's the most afraid of. Neither Batman nor Bruce Wayne are nice guys.
Batman is special precisely because of the conflict between his ethics and his anger. Superman wants to save the weak little human race. Spider-Man is just a nice guy with some guilt issues who happens to spin webs. Thor is some divine being thrown to Earth and does some nice stuff for us. Captain America is just too good to be true.
Batman is one of the most human and flawed of all the superheroes out there.
Batman walks the line between superheroism and villainy. He saves people, but he very badly wants to hurt people as well. We get a sense that, one wrong move, one wrong look, Batman would become a Joker.
This fundamental conflict between one's own goodness and evil is so rich, complex and relatable.
Batman is the man that we wish we could be and are afraid to be. He has lost his faith in law enforcement and the judicial system. Many of us have lost that faith as well. Batman struggles with the idea that he could go bad at any moment; we do as well.
And so Batman is our superhero.