Four generations, one single name: Bette Kane, the original Bat-Girl, who briefly partnered with Batwoman, Batman, and Robin. It is unknown whether in the Post Infinite Crisis continuity she ever carried this name, but later she came to be known as Flamebird, and is bound to make a huge comeback in the pages of Batwoman.
Then we have the most enduring, most recognizable, and the best incarnation of the character: Barbara Gordon, who since 1967 has partnered not only with the Dark Knight and the Boy Wonder, but also with pretty much every Super Hero in the DC Universe. This Batgirl is what all the others wish to become.
One of the top five fighters in the DC Universe and a master assassin, Cassandra Cain became Batgirl after 'No Man's Land'. After the events of Infine Crisis and One Year Later, Cassandra became a villain, but currently her status is somewhat ambiguous; we can't even tell whether she is a Batgirl anymore.
Fianlly, the most current incarnation of the character: Slowly but surely, Stephanie Brown made her way into the Bat-Family, first as Robin's partner The Spoiler, and then as the fourth Robin, Stephanie was apparently killed during 'War Games', but after being revealed that she was actually alive, the blonde teenager became the newest Batgirl.