Let me just get three things out of the way.
1) I am a big Superman fan. Make no mistake, I'm a big Batman fan too, but as a Superman fan, if there's anything I'm tired of hearing, it's that Batman can either A) totally beat Superman in a fight somehow or B) is somehow an inherently better character. I find that a lot of Superman hate is silly and hypocritical, and the idea that Batman has a chance against Superman is laughable.
2) I'm not about to suggest that what makes him special is that he's a superhero without powers. That is not a unique trait. There's a ton of street level superheroes out there who have no powers. I'm also not going to suggest that it's his intellect or his wealth and gadgets, because again, these are not unique things. Prior to Batman, you had a slew of pulp heroes with the same elements, with The Shadow being a big inspiration to Batman specifically. Finally, I'm not going to suggest that it's because he has a colorful cast of memorable villains. That's something Batman picked up from d**k Tracy. People like to pat Batman on the head for having traits that have existed before and after him in many other characters, and that really annoys me.
3) I'm not about to suggest that one version of Batman is superior to all others either. For me the 60s Adam West Batman is just as cool as Scott Snyder's Batman, or Grant Morrison's Batman. I have a soft spot for any version of Batman that harkens back to his 1939 roots as a Shadow-esque pulp hero, of course, but that's a personal treat.
NOW THEN...what makes Batman so special? I think it comes down to a few things.
For starters, Batman is basically a perfect storm of characteristics that have worked in other characters before. The wealthy playboy who masquerades as a vigilante had been done before. But not necessarily with the kind of rogue's gallery he ended up with, which again is a borrowed idea from d**k Tracy. On top of those things, if you really think about it, he exhibits some traits that are almost villainous. Think of how he seemed in 1939. He wore a darker, more colorless outfit than most such characters. He had a secret underground lair underneath a spooky mansion sitting on a cliff on the outskirts of town. Even today you see these traits utilized for villains occasionally. Now consider that he's a detective. And a martial artist with a history of training overseas (something that also comes from the Shadow).
Batman is a combination of ideas that have been fit together just perfectly. Toss in that effective origin of his and you have quite a compelling character.
But, I'd say the Shadow is a compelling character too, so there must be more to it.
For one thing, it's worth mentioning that Batman was not always big kahuna. Superman wasn't either when Captain Marvel (now Shazam, sigh) came on the scene. Superman didn't get big until after Captain Marvel was sued out of existence, but Batman? Batman got big because of the 60s TV show. Don't let anyone fool you. That's what catapulted the character into prominent mainstream media attention. He had a couple serials before then, sure. So did Superman. So did Captain Marvel. So did a ton of characters. He wasn't exactly special. But then, that show came along, and it was a smash success. People like to imagine that it almost killed the character but that's a lie too often told. People also like to imagine that it's responsible for making Batman silly and goofy, another lie.
But I digress. After that show, things died down a little bit, but the launch sequence had already started, and even in the comics Batman was taking off. In the 70s you had the awesome stuff from Denny O'Neil and Neal Adams. You had Steve Englehart and his awesome stuff. Frank Miller came along with The Dark Knight Returns and, in my opinion the far superior, Batman: Year One. Then Tim Burton's movie came out and it was a firestorm of hype. I don't think any film since has had the hype that film had, not even The Dark Knight. And through the 90s there were a few Batman movies. And then starting in 2005 there were three more. And between 1992 and now, Batman has had four animated series, not counting prominent roles in two Justice League shows (which for some reason everyone likes to remember as one show) and a Young Justice show.
Toss in all the video games and toys and my next point is pretty clear: part of Batman's popularity comes from the simple fact that over the years, there has been a lot of Batman stuff for people of all ages to absorb and enjoy. He became a marketing juggernaut that simply cannot be knocked down.
However, there's still two other factors. None of those things would be as popular if not for one simple truth: there are a lot of good Batman stories out there. People with a lot of talent and a lot of vision have told some pretty amazing Batman tales over the years. I named a few of them up there, but there's even more than those. Many more. It's not just that there's been a lot of Batman. There's been a lot of good Batman too. A lot of Batman stories that really resonate with people, that mean something to people. Some Batman stories are so good they're practically literary graphic novels.
And why is that? If anything has been shown over the years, it's that Batman is versatile. The 60s Adam West Batman is just as much a true depiction of the character as Christopher Nolan's films, or the 90s animated series. It doesn't matter what kind of superhero stories you like, whether they're dark and gritty and complex, or simple and light and fun, or any other kind in between. There's a version of Batman for you, and for everyone.
Have there been clunkers? Sure. Batman & Robin sucked, we all know that. Even though I'm sure plenty of little kids may have liked it, and maybe a few adults too. Who knows? Either way, Batman has survived crap like that for all the reasons I mentioned above.
Batman just...works somehow. Even I'm not sure exactly how all those pieces managed to fit together so well, but they have. There's no going back. Bob Kane and the incredibly underrated Bill Finger (who is responsible for most of the ingredients of Batman despite getting none of the credit he deserves) changed things. Forever. There's no going back.
Batman's just an awesome character. This year marks his 75th anniversary. I bet the next 75 years are going to be a good run for him too.
Now if I could just get people to appreciate Superman more... sweatdrop