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). In it, Joker attempted to relive his time as the Red Hood after Batman and Robin held a presentation at a college on how to do detective sleuthing, although a gardener had managed to ambush him and stole his costume. Batman eventually stopped the Red Hood, although he deduced the man who used the costume was an imposter. Joker then revealed that he had been the Red Hood of the past, and explained that he had intended to steal over a million dollars from his boss at the Ace Chemicals Plant and then retire, but then Batman interfered. He ultimately survived falling into the vat via a special apparatus in his Red Hood that allowed him to retain oxygen and swam all the way back. When he got back to his home, and removed his Red Hood, he noticed that his body had been discolored from exposure to the chemicals, and decided to adopt the identity of The Joker.
The Red Hood's origin was later revisited in Batman: The Killing Joke. Like in the above comic, the Joker was the Red Hood during that time, although it was revealed that he was not the sole holder of the identity (and in fact, was simply the latest to hold the title), and he had more sympathetic and tragic motivations for becoming it: A man named Jack had a pregnant wife, Jeannie, and they lived in a rundown tenement on the far side of Gotham City. In a desperate effort for money to provide for his family due to his dream job of being a comedian that turned out poorly, Jack resorted to a life of crime and placed himself in mob affairs. One night, the mob hired two men to kill Jack. The men let Jack do their wishes under the alias The 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 shot the other criminals dead. The engineer tried to flee, but Batman appeared and cornered him on the plant's catwalk. Terrified, he jumped off the catwalk into the chemical basin to escape, swam to freedom, and survived because of a special breathing apparatus that was built into the helmet. The toxins in the vat permanently and grotesquely disfigured him, and turned his hair green, his skin white, and his lips red. Upon discovering that, Jack went insane, and became The Joker. On a related note, how Jack discovered the deformity was altered from the above comic, as he was shown to have discovered it almost immediately afterwards, and was even shown to be in agony from exposure to the chemicals as well as confused as to why he burned and itched all over before he noticed his new appearance. After they believed Jack to be dead, the mob then hired a corrupt police officer to take his wife out. He succeeded, and Jack's wife, along with their unborn baby, burned to death in an alleged "accidental" electric fire. That all occured as Edward Nashton watched in horror. Jack became The Joker and formed a brief alliance with Nashton to search for the corrupt cop who killed his wife. Strangely enough, Joker himself was reluctant to admit that this iteration of his story was definitive, and stated: "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 that the Joker ever took up the identity of the Red Hood again was after his latest scheme had been thwarted. After he sought to restore confidence in his abilities, the Joker donned the domed helmet and tuxedo of the Red Hood again and committed 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, was murdered by the original Red Hood or more commonly known as The Joker. When Superboy-Prime altered reality, Jason returned from the grave. After he was enraged that Batman didn't avenge him, Jason located the Joker, stole the Red Hood outfit (but not before he beat him brutally), and took up the mantle of the second Red Hood. He used aggression and fear to control strong gangs and the mob, and supposedly saved the city. Jason was then found out of his activities by Batman. They argued over their theories of crime-fighting through combat, and Jason eventually showed his face and confirmed Batman's suspicion.
In Other Media
The Red Hood appeared in the Batman: The Brave and the Bold episode, "Deep Cover for Batman!" and was voiced by Jeff Bennett. That version was a heroic alternate reality version of The Joker. Like the Joker, that Red Hood was disfigured after he fell 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 was said to have been "bent, but not broken." Red Hood then tried to rally his world's heroes (alternate versions of the villains from the "normal" universe) against the Injustice Syndicate, but they were defeated. Red Hood escaped and tried to use a device to recruit help from an alternate Earth (Batman's Universe), but he was captured by the Syndicate.
After Batman was attacked by his alternate-reality doppelganger, Owlman (sent to Batman's universe on a reconnaissance mission), he journeyed to Red Hood's dimension. During scenes in that episode, the Red Hood's face was shown, but was in shadow, which showed a bit of green hair, white skin, and a wide grin that clearly resembled the Joker. After the heroes were freed and the villains were defeated, Red Hood thanked Batman and hoped that his counterpart could return the favor. Sure enough, in the next episode, "Game Over for Owlman," Batman was forced to team-up with the Joker in order to defeat Owlman, who impersonated Batman and ruined his reputation in his absence.
That Red Hood was shown to be an extremely capable fighter, and was able to hold his own against multiple members of the Injustice Syndicate. In addition, he wielded projectile weapons that were shaped like spades (a reference to his alternate universe counterpart's playing card motif).
Red Hood appeared in the DC Animated Film, Batman: Under the Red Hood, that was based on the graphic novel. Here, it was mentioned that several criminals had worn the guise of the Red Hood besides The Joker and Jason Todd. Jason's Red Hood was voiced by Jensen Ackles. A flashback also implied that the Joker (or rather, the man who became the Joker) was forced into committing the heist.
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.
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.