During a conversation of Batman's creators Bob Kane and Bill Finger, Bill was bugged by the fact "that Batman didn't have anyone to talk to" and decided to make Robin Batman's "Watson". Despite popular belief, Robin is named after Robin Hood not the bird (although some recent appearances of Robin contradict this).
Dick was the first Robin and has proven to be the most popular. Dick was the son of a family of acrobats, but Boss Zucco kill them for not paying him money (he sabotaged the ropes). Batman adopted the boy and trained him to be his ward. They took down Zucco. In the 1980's, Robin flew the nest and became Nightwing. More recently after the apparent death of Batman, Dick has become the newest version of Batman.
Jason was an overly-cheery carbon copy of the first Robin right down to his origin (although Todd's parents were killed by Killer Croc). After Crisis on Infinite Earths, Jason's origin was rewritten so he was an orphan that Batman met when he trying to steal the tires off the Batmobile. After helping stop bank robbers, Batman took him under his wing. However, this version was unpopular. The fans got to decide if Jason lived or died! By a slim amount of votes, Jason died. In A Death in the Family, he was beaten to death by a crowbar. However, he somehow survived and became the Red Hood II. After Infinite Crisis, Jason became Nightwing, but later 86ed it for the alias Red Robin. As Red Hood (again), he tried (and failed) to ruin Batman's rep.
Tim was a Batman and Robin fan boy, who managed to put 2 and 2 together and figured out who Batman and Robin really were. He managed to convince Batman to make him the 3rd Robin. He was better received than Jason. He joined Young Justice and later reformed the Teen Titans. After the events of Final Crisis and Battle For The Cowl, Tim became Red Robin.
So far, she is the only cannon female Robin. She was Tim's girlfriends, who first used the alias the Spoiler. However, she took up the mantle of Robin, when Tim resigned. She was fired by Batman for disobeying him. She tried to prove her worth, but accidentally caused a gang war, that she seemly died in (it turned out Dr. Leslie Thompkins faked it). Stephanie is now the newest Batgirl.
He was illegitimate son of Talia al Ghul and Batman. Batman didn't know about Damian, until Talia gave him to Bats to take care of. Having been trained by the League of Assassins, Damian is violent and lacks morality and discipline. He was very murderous, leading him to clash with his father (who swore to never take a life). In Battle for the Cowl, Alfred made him Robin and got him to save Tim. Now, Damian is Dick's (who is the new Batman) sidekick and Dick thinks he still needs training.