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The Trusty Guide to the Batfamily
This is Bruce Wayne. He has a lot of kids.
The first kid is Dick Grayson. Bruce adopted Dick when he was 10-12 years old after Dick's parents were murdered during an acrobat stunt at the circus. Dick was trained as Batman's sidekick, Robin, and legally adopted by Bruce. He is legally his son.
Dick grew up to be a rebellious teen (the punk) and left Gotham to become his own vigilante hero. He became Nightwing.
While Dick was away being an angsty teen/young adult, Bruce found a kid trying to steal the tires off the Batmobile and he promptly took him home. We are to assume he was an orphan, but for all we know, he could have been kidnapped.
So now, Dick is Nightwing, and Jason is the new Robin.
So far so good right? Batman (Bruce), Nightwing (Dick), Robin (Jason).
Well Jason grew up to be a little punk too and was a very rebellious sidekick who ended up going rogue on a mission and got beaten with a crowbar and blown up by the Joker.
Bye, Jason.
Well, Batman's former teacher, Ra's Al Ghul-now a criminal overlord, took pity on Bruce and dumped Jason in a magic pit called the Lazarus pit which can bring people back to life, but if you're not an Al Ghul, you get brought back to life slightly insane.
So Jason's alive again, and he's very upset with Bruce for not killing the Joker. So he pulls a big Terra/Faith and comes back as a villain (later Anti-hero) the Red Hood.
Meanwhile, while Jason was "dead", Bruce adopted another kid, Tim. Tim started to train as Robin. So now there's Dick (Nightwing), Jason (Red Hood), and Tim (Robin)
While Tim's Robin, he ends up meeting another vigilante hero going by "Spoiler".
Spoiler is Stephanie Brown and they end up dating and Stephanie becomes the first girl Robin.
But she is promptly fired by Batman because she cheats on Tim and becomes pregnant. Don't feel bad for Tim. He's spending time with his "special" friend, Konner Kent aka Superboy. So Stephanie disappears for nine months, gives her baby up for adoption, and returns as Batgirl.
So now we have Dick (Nightwing), Jason (Red Hood), Tim (Robin), and Stephanie (Batgirl). Well, Bruce gets a nice surprise around the world that he has a son. He might have got drunk with Ra's Al Ghul's daughter ten years ago, and they're shipping him his very own flesh and blood spawn. Enter Damian Wayne.
Damian has been raised Heero Yuy style and has zero social skills. So Bruce does the one thing any sane person would do with all of these kid problems. He fakes his death to keep them all "safe". Just as he had started to train Damian as a Robin, which ticked Tim off b/c he was the baby and now he's not, so Tim gets the creative idea to become "Red Robin" b/c that's so different.
So while the kids think Bruce is dead, Dick becomes Batman so no one will suspect Bruce Wayne was Batman. He continues training Damian as Robin, but Damian is also becoming super attached to him.
Bruce comes back and so it's back to Bruce being Batman, Dick being Nightwing, Jason being Red Hood, Stephanie being Batgirl, Tim being Red Robin, and Damian being Robin. Barbara Gordon is Oracle (former Batgirl until Joker shot her in the spine).
And then about a year later, Selina Kyle pops up with a surprise.
Helena doesn't really have much of a story. Her timeline skipped ahead to where she's like ten, but she started off in her own universe as an adult.
So that's just about everyone. Hope that helped you a little bit.