Batman? Yeah, he doesn't have much of a fan base…
Okay, fine. Few pop-cultural characters have as many fans as this guy. They rarely congregate in formal clubs like, say, Oz fans do, and while you can find plenty of fan pages dedicated to him (like this one), it doesn't have the codified "dues and newsletters" thing that a lot of fan bases have.
In part that's because it's one of the biggest fan bases in the world. Any kid who ever put on a cape and ran around his or her back yard can be counted as a Batman fan, and there's a heck of a lot of kids in the world. (Girls don't get left out of the fun either, not with the likes of Batgirl and Catwoman fighting alongside the Caped Crusader.) So while few of the fans are as hardcore dedicated to Bats as other fan bases are dedicated to their heroes, it's hard to find somebody who doesn't know all the basics about this guy.
That's because he's gotten bigger than most movie characters. Kids grow up reading his adventures in the comics, or watching them on TV. Then they become parents and show their kids the same thing, who grow up and show it to their kids, and so on. Suddenly, everyone knows who Bruce Wayne is, why he does what he does and what things like the Batmobile and the Bat Signal mean. Adam West gives way to Michael Keaton, who gives way to Christian Bale, who gives way to Ben Affleck, but they're all still Batman, and eventually it doesn't matter which story you see first. Sooner or later, everyone becomes a Batman fan.
These days, the fans show up most prominently at science fiction and comic book conventions, where cosplayers can blow our socks off with their dedication and authenticity. They vary their looks a lot: this guy and this other guy clearly have different inspirations, but Nolan's particular visual stamp definitely shows up a lot. It's hard to deny that this costume stems straight from Nolan's movies.
The Joker gets into the act, too. If you went to the famous San Diego Comic Con in 2008 or 2009, you couldn't sneeze without knocking over someone dressed like this or this or this. All of these folks go back to the point we made above: dressing up like heroes and villains is a way of showing how much they mean to us.
Believe it or not, that's a very old tradition: harkening to the earliest days of civilization when tribesmen would dress up like the gods and monsters of their oral stories as a way of embodying the characteristics they represented. The Greeks wore masks in dramas to depict gods and heroes, while Italian festivals such as the Carnival of Venice featured people dressing up as different characters to join in the festivities. (In fact, the Joker's girlfriend in the comics, Harley Quinn, is a riff on the harlequin, a popular Italian character whom people dressed up as.)
Cosplayers and similar Batman fans are simply doing what people have done from time immemorial. They dress up to show their appreciation for the character, but also to demonstrate some of his or her qualities… and maybe tap into some of the magic that makes them so beloved in the first place. Hey, it beats sacrificing goats, and usually gets them more clicks on YouTube in the bargain.