Do I look like Batman to you? Rethink your answer before I fill you up with lead.
— Red Hood to Black Mask Thug
The Red Hood was the alias of a particularly infamous criminal active during Batman's early years as the Caped Crusader. As indicated by the name, the criminal wore a red bowl-shaped helmet as a means of covering his identity, although it is strongly implied that the person wearing the disguise was always different. The Red Hood is most notorious as being the former identity of The Joker prior to his encounter with Batman.
Since Jason Todd changed the Arkham Knight persona, he was able to lay his long-rooted vengeance against Batman to rest, and began to heal the damage that Joker inflicted on him at Arkham Asylum. It was during this recovery that the Red Hood was born, and is implied to have become the permanent holder of the identity. However, while he was once again aligned with his former mentor in wanting to rid Gotham City of its criminal contingent, Todd's willingness to take lives will forever keep them at odds.
Batman: Arkham Origins
Outside of The Joker's presence in the plot (and his psychiatric evaluation), the Red Hood was alluded to on the crime-board in the Batcave, which was listed under the second row, as well as on a separate billboard. According to the information, in addition to the Red Hood's role in the Ace Chemicals Heist, the Red Hood also had undergone several daring heists and major scores. His schemes were detrimental enough to Gotham City for Mayor Hill to make the apprehension of Red Hood a high priority for the GCPD. He also struck again sometime later by taking $300K worth of money, which lead to Kale to promise to unmask Red Hood, although some notes on the crime-board indicated that Batman wasn't sure if that was actually Red Hood who did it. His actions ended up costing Gotham $1,000,000, at least at the time that the report was made. Commissioner Loeb later claimed that Red Hood was "run out of the city." Batman also speculated whether Red Hood was in fact several criminals who used the moniker. Sometime after that, the Red Hood case went cold.
Batman: Arkham Knight
When Jason Todd was unmasked as the Arkham Knight after his confrontation with Batman, he began to realize that Bruce was truly sorry for what had happened to him with the Joker. Jason tracked Batman after he surrendered to Scarecrow's demands. Jason, as the Red Hood, disarmed Scarecrow just as he prepared to kill Bruce and then freed him with a well-aimed shot to his restraints, and allowed the latter to subdue Scarecrow with his own Fear Toxin.
Later on, Red Hood began his work on taking down Roman Sionis, Black Mask; and interrogated several of his lower class henchmen. Red Hood managed to take down a gun shipment, learned that Sionis was holed up in his office and started on his way there. Arriving, he easily dispatched the goons and cornered Black Mask. Sionis continuously pleaded with Red Hood for his life, and offered him drugs, money, weapons, and even promised to leave Gotham and go anywhere that he wanted. Red Hood's response: "How about you go to Hell!" Before he left, Jason added: "Say hi to Joker for me." before he set off into the streets.