The Red Hood was the original Joker origin story. Similar to the Jack Nicholson origin in Batman, The Red Hood was cornered by Batman, dove into a vat of chemicals to avoid capture, and came out of it with green hair, white skin, and went completely insane.
In The Killing Joke, Alan Moore changed the story a little (major Killing Joke spoilers ahead...). First, the future Joker wasn't the only one to wear the Red Hood. A couple of thieves would recruit an inside man for each job and have him wear the hood, so that if they were caught people would assume the Red Hood was the mastermind. It worked perfectly, Batman walks up to the first time criminal under the hood and says "we meet again," causing a terrified future Joker to flee into a vat of chemicals. The best touch of The Killing Joke was that after all the flashbacks supposedly telling his origin, the Joker notes that every time he remembers the story it's different ("If I'm going to have a past, I prefer it to be multiple choice!"), an element of the character that got incorporated wonderfully into The Dark Knight.
Another Red Hood appeared more recently, in one of the stupidest retcons ever. Jason Todd was the second Robin. In a controversial storyline, readers were asked to call a 1-900 number to vote whether or not Jason, who had grown somewhat unpopular, should be killed off. In a close vote, readers gave Robin a thumbs down, and the Joker beat him to death with a crowbar, and then I think blew up the building he was in. It was kind of an ugly moment in Batman history, but the third Robin turned out to be a much better character (I think), so it was best to move on and forget about it. But then about 15 years later, and there's no way to put this that doesn't sound stupid, Superboy Prime punched the walls of reality, causing a ripple effect that changed all sorts of events in the DC universe, including Jason Todd being suddenly alive again. He put on the Red Hood and it brought back unpleasant memories of the 1-900 thing and, in my opinion, was generally pretty dumb.
Pros: The Killing Joke was great.
Cons: The Joker is best without an origin (or with multiple choice origins), and bringing in the first Red Hood would require not only a new Joker story without Heath Ledger, but flashing back to an origin. For the other Red Hood, we likely won't be seeing any Robins any time soon, so that's not going to happen.
How I'd do it: I guess if you had to flash back to some time around Batman Begins, you could have Batman beat down on a criminal in a red hood as a little easter egg for fans, but anything more than that would be a mistake.
Casting Guesses: I'm going to say none.
Verdict: Not going to happen. But given the significance of the character I thought he was worth mentioning here. Plus, that's a pretty cool image up top, stolen from Matt Wagner.
After The Scarecrow, Ra's Al Ghul, The Joker, and Two-Face, who should be Batman's next enemy in the Batman Begins/The Dark Knight series? Well, as of now we know that Tom Hardy will play Bane and Anne Hathaway will play Catwoman.
Check out the master list for all the villains covered. I've pretty much run out by now. Obviously, there are more Batman villains, but not any of significance that I know of. As of February 2009, no one is confirmed as a villain. Not the Johnny Depp Riddler, not Phillip Seymour Hoffman Penguin, nobody.