Red Hood appeared in 1951 comics called Detective Comics. He's actually Joker in 1951 comics, but in 2006 he's none other than Jason Todd who was the former Robin killed by the Joker in Batman: A Death in the Family, is revealed to have been resurrected by Ra's al Ghul via the Lazarus Pit. But the pit changes him and his emotions and he becomes the new Red Hood. His debut culminates in a fateful confrontation with those he feels have wronged him. He beats the Joker with a crowbar (mirroring the way the Joker had tortured him before killing him with a bomb) and later kidnaps him. The new Red Hood assumes control over various gangs in Gotham City and starts a one-man war against Black Mask's criminal empire. He actively tries to cleanse the city of corruption, such as the illegal drug trade and gang violence, but in a violent, antiheroic way. He eventually comes to blows against Batman and other heroes, including Nightwing, the new Robin (Tim Drake), Onyx, and Green Arrow.
In the second story arc of Batman and Robin by Grant Morrison and Philip Tan, Jason retakes the Red Hood mantle. With the goal of making the very concept of Batman obsolete, he puts a lot of effort into public relations: he drastically alters his Red Hood costume to look more like a traditional superhero outfit, and recruits his own sidekick known as Scarlet. In their war on crime Red Hood and Scarlet freely kill criminals, villains and anyone who gets in their way, even the police. After all of Red Hood's killings he leaves behind a calling card which states "let the punishment fit the crime". He describes his vendetta against Dick Grayson as "the revenge of one crazy man in a mask on another crazy man in a mask".
After Barry Allen's involuntary tampering with the continuum, Red Hood is shown, along with Arsenal and Starfire, as one of the Outlaws, a close knit group of anti-heroes. Still not above killing, and still angry at the world, Jason has now reverted to the street clothes costume, forgoing his feud with Batman for stealthier, more cloak and dagger missions. Eventually, Jason and Batman reconcile and call a truce between them.