Note: I focused on the darker renditions of Batman, not the candy-coated and plastic-nippled Batman.
Batman is my favorite superhero.
To understand the essence of the Batman mythos, we need to understand Batman's beginnings. As a child, he wasn't that special, just another spoiled rich boy. His parents' murder changed all that. Not only did he have a reason to take vengeance on the criminals -- vengeance, not goody-goodness -- but he also had a reason to be afraid.
I also quite 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 the most: a criminal.
Batman is half-superhero, half-villain.
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 the superheroes.
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 faith in the 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.
Batman is our superhero.