So, it's been said that the three main Robins, (Dick, Tim and Jason) are all reflective of the three main parts of Batman's personality, his compassion, his intelligence and his anger.
Personally, i kinda think that was a coincidence that sounded really cool, so DC just runs with it now, but anyway.
Jason basically started out as a carbon copy of Dick (literally, circus and everything) after Dick left, got his Nightwing on and started Teen Titaning. This wasn't an all that popular move with Jason's character, so new writers were introduced that re-make Jason to have a bad-boy attitude. If the Robins were a boy band, he'd be the one with the rebel hair cut and the goatee. His new back story re-wrote him as a street kid with a good heart that Batman found stealing his tires one night...off the bat mobile...kid had balls. Of course, as Batman is, when ever he sees an orphan boy he's just GOT to have it, so he took in Jason and trained him to be the new Robin.
Jason had a lot of anger though. He was kind of the anti-dick (why does that sound dirty?) where Dick would follow Batman's orders without hesitation, Jason had a tendency to play by his own rules, letting his anger get the best of him. This anger reflected Bruce's, which he didn't exactly like as it brought out a weakness in Batman. The two had conflicts but still cared for each other and yadda yadda yadda.
Of course, Jason's most famous moment was his death in the 80's. Fans were able to call in and have their say as to whether Jason survived a crowbar/bomb attack from the Joker or not. The fans chose the latter. It was a sad choice at the time, but in the long run, kinda cool. It added a new dynamic to Batman, helping mold the darker and broodier Batman we know and love today, paving plot road for Tim Drake later, but that's another story. On top of that, as comic book characters do, Jason rose from the dead later when Superboy Prime...punched...*sigh* reality. Even though Jason wasn't that much of a fan fav when he was alive the first time, when he came back as Red Hood, that was another story. He's probably one of the most popular new non-classic Batman villains to pop up since Harley Quinn and Black Mask. He's not even that much of a villain, more an anti-hero. He's pretty much doing what his character was doing before he died, using his anger to fight crime, playing by his own rules and exposing Batman's weaknesses. That and he got wicked sexy