Heartbroken Badass: A non-romantic version. Jason's death and subsequent downfall into villainy left him broken and angry at himself for not being able to save Jason and leaving him with the wish that he could have done more for him if he had the chance.
A Million Is a Statistic: The crux of Jason's grievance with Batman. Despite the "entire graveyards" worth of people that the Joker has killed, Jason was hoping he'd be special, that he'd be the one Batman actually avenged.
Anti-Villain: His first act upon assuming control of the various gangs is to make children untouchable to the drug trade, and he only ever targets criminals in the course of his various activities. As he tells Batman during their first real chat, he's actively attempting to control the criminal element of Gotham, but we don't find out why until the climax.
Asskicking Equals Authority: He sets himself up as the drug kingpin in Gotham and rival to Black Mask by showing up at a meeting of the "heads" of the organizations under Mask's thumb and giving them a bag full of the heads of their lieutenants.
Came Back Wrong: According to Ra's Al Ghul. The man himself throws it back in Batman's face, suggesting his current activity is somehow easier to digest if the Lazarus Pit "turned [him] rabid".
Cold Sniper: He kills three people from a significant distance without touching Batman or Nightwing standing to either side of them.
Combat Pragmatist: Though he has plenty of gadgets up his sleeves, he's not afraid to use anything in the environment to his advantage in a fight, and, as noted under The Unfettered, is willing to use any methods necessary to gain an advantage.
Complexity Addiction: His ultimate plan to get Batman to realize that sometimes Murder Is the Best Solution, specifically regarding irredeemable people like The Joker. Instead of just finishing The Joker off himself and using that as the ultimate symbol of his new pragmatic approach being more effective than Batman's idealistic approach, he goes to extra trouble to get Batman to abandon his Thou Shalt Not Kill rule and finish The Joker off.
The Heavy: The main antagonist and the one who drives the plot along for the majority of the film.
Irony: There's the fact that his actions are based around heroism, but are so brutal that he is a firm villain. But there is also a bit of symbolic irony in a bat family member becoming a new Red Hood, with the original being the original villain alias of who else but The Joker.
Knife Nut: After the guns, before the fists, he has a knife that can cut Batman's cables.
Knight Templar: Believes that everything he has done, every life he has taken, has been in the name of bringing order and stability to Gotham.
[To Batman] "Yes. You can't stop crime. That's what you never understood. I'm controlling it. You wanna rule them by fear. But what do you do with the ones who aren't afraid? I'm doing what you won't, I'm taking them out!"
Woobie, Destroyer of Worlds:He was once the partner of Batman and a former juvenile delinquent whose hard background, brutal death at the hands of the Joker, and subsequent resurrection drove him over the edge to become the murderous Knight Templar Red Hood.
Sadist: He takes immense joy out of the suffering he inflicts on others.
The Sociopath: He has absolutely no regard for human life, including his own.
Would Hurt a Child: His first scene shows the Joker beating a teenage boy nearly to death, and then finishing the job with a bomb.
"Now, that was rude. The first boy blunder had some manners. I suppose I'm going to have to teach you a lesson so you can better follow in his footsteps. (beat) Nah, I'm just gonna keep beating you with this crowbar."
Big Bad Wannabe: Prior to its theft by Red Hood, AMAZO was to be Mask's ticket into international arms deals, a step-up from his current "bottom-feeding garbage" of small scale drug and weapons trafficking. Later, after a direct attack from Red Hood, Mask decides to get out of the "cross-fire" by breaking the Joker out of Arkham, which goes about as well as could be expected.
Ms. Li: Understood. We'll be taking further precautions at every transaction.
Mask: Screw that! Time he learned that this is a contact sport. We're going on offense. Rough up his business; something big! Something loud! When he shows up to shut us down, have a party waiting for him! And when I say "party" I actually mean a 'whole lot of people who are gonna KILL HIM!
Affably Evil: He states that he never resorts to violence or killing unless it's for a "greater cause" and is deeply remorseful that his hiring of the Joker was ultimately what lead to Jason's murder; his attempt to fix things via Lazarus Pit resurrection is what sets off the main plot.
The Atoner: After Jason's death, Ra's chooses not to oppose Batman again, and tries to resurrect Jason to try and repair the damage he inadvertantly caused.
My God, What Have I Done?: After his actions cause Jason's death, and again, after hearing of Jason's actions as the Red Hood, following Ra's raising him from the dead.
Unwitting Instigator of Doom: Twice. First, when his hiring of the Joker was ultimately what lead to Jason's murder. Second, when he did try to resurrect Jason to try and repair the damage he inadvertantly caused, but his actions only made things worse.