Nightwing: Richard “Dick” Grayson, ward of Bruce Wayne, former circus acrobat whose entire family died in front of him. Currently guardian of Bludhaven and connected to their police department. Was the 1st Robin.
Oracle: Barbara “Babs” Gordon, daughter of Police Commissioner Jim Gordon, originally studied to be a librarian but ended up fighting crime as the 1st Batgirl instead. Left paralyzed from the waist down in an attack by the Joker, Babs reinvented herself as Oracle, the information broker for all of Bat-clan as well as the JLA.
Red Hood: Jason “Jay” Todd, former ward of Bruce Wayne, was living on the streets when he tried to steal the tires off the Batmobile and was taken in by Bruce. Presumed dead several years ago after an encounter with the Joker and a warehouse fire. Seems that everyone was wrong. Was the 2nd Robin.
Red Robin: Timothy “Tim” Drake, brilliant son of a well-off family who discovered Bruce’s and Dick’s identities on his own. Decided to join because Batman needs a Robin after Batman's behavior after the death of Jay. Was the 3rd Robin.
Batgirl: Stephanie “Steph” Brown, daughter of the villain Cluemaster. Got into fighting crime under the name Spoiler, as she mainly spoiled her father’s plans, but met Tim when he was Robin and fell complicatedly in love with him. Was briefly the 4th Robin, under Batman’s tutelage, but is now Batgirl, trained by Oracle. (ETA: After double-checking our canon, depending on how you count, Steph is either Batgirl III or Batgirl IV, depending on whether you count Helena Bertinelli's brief--and non-Babs-approved--stint as Batgirl.)
Wonder Girl: Cassandra “Cassie” Sandsmark, daughter of archaeologist Helena Sandsmark. She is literally a demigoddess: her father is Zeus. Yep, really. And you thought you had issues with your parents. She doesn't talk much about that part of her life. Idolized Wonder Woman as a child, and was trained by the Amazon Princess to be the 2nd Wonder Girl.
Black Canary II: Dinah Lance, daughter of the original Black Canary. She is a metahuman, possessing a 'canary cry' sonic scream that's bad news for anyone in earshot--even Kryptonians. Formerly the love interest of Green Arrow (Oliver Queen) and 'den mother', as she calls it, to his brood, she is also a founder of the Birds of Prey. Currently the Chairwoman of the Justice League of America, Dinah is held in high respect by the other heroes.
Huntress: Helena Bertinelli, daughter of the infamous Gotham City mob family, witnessed the deaths of all her immediate relatives at a very young age. Originally vowed to avenge the murders, becoming a costumed vigilante, but over time changed her focus to the root cause and now seeks to end organized crime. Worked with Batman very briefly under the Batgirl mantle (without Babs' approval), but fell into disfavor for her anger and violence, and was fired from the role. Now operates as Huntress under the aegis of Oracle's Birds of Prey, and walks a fine line between vengeance and justice.