I have always loved the Pulp Noir Hard Boiled Detective genre, and Batman has that in spades. I love the old timey gangster genre, Batman has that. Batman is psychologically fascinating to me, his obsessiveness which leads him to perfect his body, physically and mentally, and prepare for any scenarios (one reason I'm loving Scott Snyder's recent run is that he finds great ways to challenge Batman.
Batman is the World's Greatest Detective, he's Sherlock Holmes blended with James Bond (suave, expert in hand to hand combat, awesome gadgets and vehicles) and he's a ninja:
He has the greatest Rogue's Gallery of any hero, imo, each villain unique and colorful with their own paticular psychological obsession.
Batman has so much versatility, his stories can be a mystery, a sci fi action adventure, even a horror themed story. He can be down on the streets, taking on the mob, street dealers, low level thieves and punks, and just as easily be pit against a deadly assassin, a powerful alien, or a God level meta human. His stories can have a sense of humor or be dead serious, dark and gritty. Just the environment of Gotham itself appeals to me in terms of what I like in fiction. And the supporting characters, Gordon, Barbara, Dick, Tim, Alfred, etc. are all great in their own rights.
Those are some of the main reasons Batman is my favorite comic hero. Also, he has a really big heart when you get down to it, and he's very flawed and relatable.