Robin the first: Dick Grayson. This is the Robin most people think of when they think of Robin, this is the kid raised in a circus who watched his parents fall to their death and was adopted by Bruce Wayne. Dick is a acrobat, one of the best in the world. The name 'Robin' came from his mother's pet name for him - he 'flew like a Robin' while part of the Flying Graysons trapeze act. Then he was Robin. Then he wasn't. Disagreements with Batman and the desire to be his own hero lead Dick to forge the persona Nightwing (a tribute to the superhero mythology of Superman's people), and he left Gotham. At one point he donned the Batman costume while Bruce Wayne was recovering from a spinal injury.
Robin the second: Jason Todd. Due to the reboot DC Comics went through while Jason Todd was in costume, he has two backstories. The first is nearly identity to Dick Grayson's - a circus kid, parents killed tragically, adopted by Bruce Wayne. He was Dick v2.0, and it didn't go over well. After the reboot he was a street kid who tried to lift the tires off the Batmobile. Bruce eventually brought him in as Robin when he proved he could help with crime solving, but Jason was rash and impulsive and eventually got beaten to death with a crowbar by the Joker. His costume hangs in a lit glass case in the Batcave, because Batman is creepy like that.
Robin the third: Tim Drake. Tim was at the circus the day Dick's parents died, and met Dick before the act. He watched Dick perform the quadruple somersault (a feat only three people in the DC world can do), and then watched Dick's parents fall to their death. He watched Batman come up and take Dick under his arm, followed the adoption by Bruce Wayne in the newspaper. And one day, as a kid, he saw footage of Robin in action - performing the quadruple somersault. He gathered evidence and proof and photographs that, yes, Dick Grayson was Robin, and Bruce Wayne was Batman. Then Dick wasn't Robin, and Bruce Wayne adopted a new kid, and there was a new Robin. (I can't imagine how no one else figured this out.) Then - oops - Jason Todd was beaten to death with a crowbar. And there was no Robin again. And Batman pretty much went insane. Tim, being a conscientious stalker, tracked down Dick Grayson and begged him to be Robin again. Tim's basic theory was that 'Batman needs a Robin'. (Translation: Batman is less totally and completely freaky insane, and hurts a lot less people. Valid point.) After much yelling all around, Tim is made Robin. He promptly travels all over the world and becomes an adolescent ninja assassin, essentially. Then he gets a better costume than the other guys. Then his Dad figures out he's Robin, and he quits for a bit, leading to..
Robin the fourth: Stephanie Brown. Stephanie's dad was basically the poor man's Riddler, calling himself the Cluemaster. Except who wants a lame supervillain for a dad? Stephanie decides as a kid that Batman's much cooler, and goes about foiling the Cluemaster's plans and making things easier for Bats. She makes herself a giant purple (eggplant!) hood-and-cape ordeal and calls herself Spoiler. Batman has Tim-Robin follow her a bit, where she explains herself, and eventually she and Tim become a couple. Because DC Comics is weird like that. When Tim has to hang up the cape, Stephanie steps in. Angst ensues. Stephanie-Robin finds some of Batman's war game plans (because Batman has plans for everything) and sets off a giant-ass gang war wherein she gets herself killed. Oops. Bye, Stephanie.
Now, DC Comics occasionally publishes 'Elseworlds' stories that don't fit into their own continuity, like Superman landing in Russia instead of Kansas, or Batman squaring off against Dracula-types. One of these stories, Dark Knight Returns, published in the 80's, had a much older Batman operating at some point in the future. Frank Miller, who wrote the book, included such things as a dead Jason Todd (complete with glass case in the Batcave) that eventually became incorporated into the real comics. It also included a female Robin named Carrie Kelly - the first time we'd seen a female Robin. Then there was a sequel. But, well, she doesn't count. And I never read the sequel.
Summary: Batman needs a Robin, and being a Robin is kind of cool until someone beats you to death.