To the outside world, she's the librarian daughter of Gotham City Police Commissioner James Gordon. To those in the know, Babs is the mama bird of the dysfunctional Bat family and the boss of pretty much everyone else ever. She was the first Batgirl and after a run-in with the Joker that left her paralyzed from the waist down, she became Oracle. As Oracle, she ran the DC Universe due to her gigs with the Justice League and the Birds of Prey. She has since regained her mobility and is back on the streets of Gotham as Batgirl, bossing people around face to face.
Let's get this out of the way first: Cass easily wears the creepiest Bat costume. But the gimp mask has a purpose! Cass is the daughter of David Cain and Lady Shiva, two premier assassins, and she was raised as the perfect fighter, responding only to body language. She could not read, write or speak. See, the mask is all poetic and whatever. Cass later learns English, gives the Batgirl mantle to her BFF Steph, and becomes the Black Bat.
She's been Spoiler, a superhero focused on spoiling the plans of the Cluemaster, her deadbeat villain father. She's been Robin, flying high over the city of Gotham. She's been Batgirl, where she came into her own and starred in one of the best Bat books ever written. Steph's had a quite depressing storyline and publication history but there are two things you must remember about this young woman: she doesn't buy into Batman's doom and gloom approach to life and she slapped Batman in the face after he came back from the dead.
Kate is a Jewish lesbian that decided to become Batwoman after being kicked out of West Point under "Don't ask, don't tell." (Isn't it nice that her cutting edge origin story is now a tiny bit outdated?) There's more to talk about with her, but let's just look at the art. Best art ever.
Bette is the younger cousin of Batwoman. As Flamebird, and later Hawkfire, she is Batwoman's go-to-girl. Guys, she kicks ass in ballet flats. Ok, not really but it's fun to pretend. Fun fact: Bette first appeared as Betty, the original Bat-Girl, in 1961. She worked with Batwoman back then too. She also constantly tried to kick it with Robin. Back when he unironically wore hot pants.
She fights crime with a slingshot and fireworks while wearing the coolest glasses. Carrie is a Robin of the future, specifically the Robin from The Dark Knight Returns. She goes toe to toe with the Joker and his accomplices and wins! Hell, she even fights Superman. Future Batman may be a big ol' grumpasaurus (just like regular, present day Batman) but Carrie makes it work.
She's so mysterious! The newest Batgirl has popped up in Batman Beyond Unlimited, a digital comic based on the turn-of-the-millennium cartoon. (Don't you miss using "the millennium?") Who is she? What does she want? No one knows but Gotham City Police Commissioner Barbara Gordon (the older woman with the Carrie-Kelleyesque glasses in the background) is gonna find out.