You might not know it from the current run of Batwoman comics, but when Kathy Kane was introduced in 1956 it was purely as a love interest of Batman, to dispel rumours about his having a homosexual relationship with the sidekick Robin. She was around for less than ten years before being written out of the continuity with a host of other “non-necessary” Bat Family members like Bat-Mite and Bat-Hound; he only temporarily needed the beard, it seems. When she was reintroduced in 2006 she became Kate Kane, a Jewish socialite lesbian who was expelled from the Army under the (now revoked) Don't Ask Don't Tell policy.
Although she wears the Bat mantle, and is in fact distantly related to Batman (his mother Martha Wayne's maiden name was Kane), they have very different approaches to the task of vigiltanism. Most notably, Batwoman carries a gun – something Batman has sworn to never do. In the Greg Rucka and J.H. Williams III series Elegy, we learned her story and discovered that her respect for Batwoman inspired her to take the mantle of the Bat, and while she isn't part of the officiall Bat Family (like Batgirl, Robin, Nightwing...) Batman doesn't try to stop her. He knows her real identity, and believes she has the right motivations – or at least, ones that aren't too far from his own.
The success of Christopher Nolan's Dark Knight trilogy has definitely been a boon to DC, reigniting interest in the caped crusader which has resulted in the critically acclaimed Arkham video game series, Dark Knight Rises slots and new additions to the mythology coming regularly. Most recently we have seen the death of Damian Wayne, Bruce's son and the latest Robin; it will be interesting to see how the grief will affect his clash with the Batwoman, and whether they will discover that they are related.
Currently in the serial, Batwoman is being forced to discover Batman's identity in order to save her father from prison and her fiance and cousin from danger, querying Bane and Black Mask in the process to find out Batman's vulnerabilities. The work has been a move from the more supernatural elements of Batwoman so far in the Religion of Crime, and the clash between the two Bat-Adults will have long-term consequences for our Batwoman as well as highlighting the differences between the two.
Trevor McCarthy has taken over the role of artist, adding a surreal element to the imagery as the story moves away from the truly bizarre and into a more comfortable Bat-Zone, and writing will be dealt with by W. Haden Blackman and J.H. Williams III. It's shaping up to be an exciting story, with DC promising that this “no-holds-barred battle” will “threaten to kill them both!”