- " Because at the end of the day what people are afraid of is the nothing of it, Bruce. The randomness. The empty center. Stare into it and try to find meaning. You'll go mad. All you can do is fear, and survive. It's the truth. That's what this gang is--you see, Bruce? I came up with the idea for it in the wake of your parents' death. In that wonderful moment--that true moment, right after. See? We wear the red hood to court the wolf, rather than hide. Eat us, we say. Eat us all. "
- ―The Red Hood
In the story Batman: The Killing Joke, the Red Hood Gang was first introduced as being a criminal operation run by gangsters who created the super-villain alias of "The Red Hood" in order to act as a red herring which would distract the authorities and Batman from pursuing the real men in charge. In a backstory to Joker which might very likely be a derranged fabrication, Joker recalls having been one of the decoys to take-up the Red Hood mantle in-order to help the gang rob his old workplace of the ACE Chemicals Plant. During the robbery, the gang was ambushed by the police and by Batman with the gangsters running the operation having been shot dead while the Red Hood was chased by Batman to a cat-walk which he fell off, falling into the vat of chemicals which supposedly dyed his hair green, bleached his skin white and gave him his signature red smile.
Years after this would have taken place, the Red Hood mantle was taken up by the ressurected Jason Todd who started conquering the criminal-underworld to act as a gangster that started a war with Gotham's then most powerful crime-lord the Black Mask. This Red Hood however had a moral-compass and used his influence over Gotham City's gangs to reduce violence and drug-trafficking.
Prime Earth (New 52)
In the New 52, during Batman's first year in Gotham the Red Hood Gang was his first big opponent. The Red Hood gang was lead by the man who would become Joker and who was already a twisted, homicidal nihilist dedicated solely to chaos. However most of the gang's members were not willing participants but instead middle-to-lower class residents of Gotham being blackmailed by the Red Hood and made to wear their own variation of the Red Hood mask and each given a code-number while the future Joker was "Red Hood One".
Red Hood commit several robberies and attempted to create a chemical cocktail at Ace Chemicals which would turn the facility into a giant chemical explosion. The explosion would have destroyed much of Gotham in the name of nihilistic chaos but the plot was foiled by Batman. Rather than surrendering, Red Hood willingly dropped himself into a boiling vat of chemicals where he was presumed dead until resurfacing later as the Joker.
During the Zero Year event in which the Riddler enlisted the help of Poison Ivy and Doctor Death to black-out Gotham, several criminals attempted to revive the Red Hood Gang. The leader of this incarnation of the gang was decapitated by Talia al Ghūl and the rest were murdered by the Joker in an explosion. One of the only survivors of this Red Hood gang was a troubled young Jason Todd.
During a psychological-interview with the Joker conducted by Prof. Hugo Strange, Joker tells a story mirroring the New Earth variation of the gang with the only change being that here the gang was run by Carmine Falcone.
In the television series Gotham, the Red Hood Gang was a minor gang of petty-criminals and bank-robbers. The gang's titular Red Hood here was derived from a crude red ski-mask which one of the members wore during a robbery where hungry for attention and admiration, he gave much of the stolen money to the hostages of the robbery to assume a Robin Hood-type persona. The members of this gang started murdering one another for the title of Red Hood and the ski-mask before they were all stopped by the GCPD.
In Batman: The Adventures Continue we are given a group somewhat analogous to the Red Hood Gang in the form of an east-side Gotham gang known as, "The Wolves". This gang themed themselves around wolves, being lead by a brutal, "Alpha" and with new members being deemed Red Hoods as an allusion to the fairy-tale Little Red Riding Hood. Red Hoods would have to wear red ski-masks and be used as precarious look-outs, often resulting in capture and death.
One of these red hoods was the older brother of a young Jason Todd who died on duty, causing Jason to take up vigilantism to fight the gang.