The first Red Hood
Main article: The Joker
The Red Hood first appeared in Detective Comics #168 "The Man Behind the Red Hood (February 1951).
A man named Jack had a pregnant wife, and they lived in a rundown tenement on the far side of Gotham City. In an effort for money, Jack resorted to a life of crime and began placing himself in mob affairs. One night, the mob hires two men to kill Jack. The men let Jack do their wishes under the alias Red Hood. His costume consisted of a large domed red helmet and a red cape. During the robbery, the plant's security guards spot the intruders and shoot the other criminals dead. The engineer tries to flee, but Batman appears and corners him on the plant's catwalk. Terrified, he jumps off the catwalk into the chemical basin to escape,and swam to freedom, surviving because of a special breathing apparatus built into the helmet. The toxins in the vat permanently and grotesquely disfigured him, turning his hair green, his skin white and his lips red. Upon discovering this, he went insane, and became the Joker. Believing Jack to be dead, the mob then hires a corrupt police officer to take his wife out. He succeeds, and Jack's wife, along with their unborn baby, burn to death in an alleged "accidental" electric fire. This all occurs as Edward Nashton watches in horror. Jack becomes Joker and forms a brief alliance with Nashton to search for the corrupt cop who killed his wife. Strangely enough, the Joker himself is reluctant to admit that this iteration of his story is definitive, stating: "Sometimes I remember it one way, sometimes another... if I'm going to have a past, I prefer it to be multiple choice!"
The only other occasion the Joker ever took up the identity of the Red Hood again was after his latest scheme had been thwarted. Seeking to restore confidence in his abilities, the Joker donned the domed helmet and tuxedo of the Red Hood again to commit a crime in Batman #450 (1990).
Four years later, the Red Hood costume finally ended up behind glass on display in the Batcave, as glimpsed in Batman #506.
The second Red Hood
Main article: Jason Todd
Jason Todd, one of the Robins, had been murdered by the original Red Hood. When Superboy-Prime alters reality, Jason returns from the grave. Enraged that Batman didn't avenge him, he locates the Joker and steals the Red Hood outfit (but not before beating him brutally), taking up the mantle of the second Red Hood. He uses aggression and fear to control strong gangs and the mob, supposedly saving the city. Jason is then found out of his activities by Batman. They argue over their theories of crime-fighting through combat, with Jason eventually showing his face and confirming Batman's suspicion.
In other media
Red Hood appears in the Batman: The Brave and the Bold episode "Deep Cover for Batman!" voiced by Jeff Bennett. This version is a heroic alternate reality version of Joker. Like the Joker, this Red Hood was disfigured after falling into a chemical vat at the Ace Chemical Plant. In his case, he was already a superhero and was actively dropped in by Owlman. However, his sanity is said to have been "Bent, but not broken." Red Hood then tries to rally his world's heroes (alternate versions of the villains from the "normal" universe) against the Injustice Syndicate, but they are defeated. Red Hood escapes and tries to use a device to recruit help from an alternate Earth (Batman's universe), but he is captured by the Syndicate.
After Batman is attacked by his alternate-reality doppelganger, Owlman (sent to Batman's universe on a reconnaissance mission), he journeys to Red Hood's dimension. During scenes in this episode, the Red Hood's face is shown, but in shadow, showing a bit of green hair and a wide grin that clearly resembles the Joker. After the heroes are freed and the villains defeated, Red Hood thanks Batman and hopes his counterpart can return the favor. Sure enough, in the next episode "Game Over for Owlman," Batman is forced to team-up with the Joker in order to defeat Owlman, who has impersonated Batman and ruined his reputation in his absence.
This Red Hood is shown to be an extremely capable fighter, able to hold his own against multiple members of the Injustice Syndicate. In addition, he wields projectile weapons shaped like spades (a reference to his alternate universe counterpart's playing card motif).
Red Hood appears in the DC animated film, Batman: Under the Red Hood, based on the graphic novel. Here it is mentioned that several criminals have worn the guise of the Red Hood besides Joker and Jason. The Jason Todd Red Hood was voiced by Jensen Ackles. A flashback also implied that the Joker (or rather, the man who became the Joker) had been forced into committing the heist.
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, listed under the 2nd 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 Hul 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, leading 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 that. His actions ended up costing Gotham City $1,000,000, at least at the time the report was made. Loeb later claimed that Red Hood was "run out of the city." Batman also speculated whether Red Hood was in fact several criminals using the moniker. Sometime after this, the Red Hood case went cold.