There have been very few characters introduced in comic books in the last 20 years who have A) a solid origin story and B) are interesting enough to hold their own title. Batwoman is one of these rare characters.
Originally introduced in the DC Comics weekly series, 52, back in 2006, Kate Kane was a wealthy socialite who started dressing up and fighting crime as Batwoman. When the character was first announced by DC, there was a lot of media attention surrounding the fact that the character is a lesbian. At the time, this reeked of a publicity stunt (though DC claimed otherwise, stating numerous times that they were trying to add some diversity to their characters by adding minorities to their roster of characters). The character was fairly forgettable, aside from her relationship with long-time DC character Renee Montoya (another long story), but then a couple of years later writer Greg Rucka and artist JH Williams III expanded on the character in Detective Comics, which featured her as the lead character for 10 issues (the first seven of which are collected in the excellent Batwoman: Elegy hardcover) and made her much more compelling. Her new backstory includes her time in the military (from which she was discharged under "Don't Ask, Don't Tell"), the death of her mother and twin sister at the hands of terrorists, and her preparations in (and reasons for) becoming Batwoman. It also explores her past relationship with Renee Montoya. All this, and she faces off against the mad villain known as "Alice" who is the head of the Religion of Crime.
Now why should I be bringing up Batwoman? Well, it just so happens she now has her own monthly series (issue #10 will be out soon) with JH Williams III and W. Haden Blackman co-writing and Williams doing the artwork for every second storyarc. The reason I bring this up is because issues #0-5 of the series are now collected (and available for sale) in Batwoman Volume 1: Hydrology which features Kate's first battles with a new enemy (The Weeping Woman) while trying to train her new sidekick (her cousin Bette, aka Flamebird) and make time for her new girlfriend Maggie Sawyer (who happens to be a cop investigating Batwoman). On top of all this, Kate has to deal with Agent Cameron Chase of the D.E.O. , an organization that watches and wants to control superheroes in the DC Universe. It's beautiful to look at, the story is compelling, and Batwoman is one of the best characters to come out of comic books in the last couple of decades.
So do yourself a favor and pick up Batwoman: Elegy and then Batwoman: Hydrology. If you dig those, think about checking out the monthly Batwoman comic book series for more on Kate Kane and her crime-fighting alter-ego.