After Dick Grayson's parents were murdered by the mob, Batman trained him to fight crime as 'Robin'. When Dick reached adulthood, he assumed his own independent identity - Nightwing.
Jason Todd (aka Red Hood) became the second Robin. Caught trying to steal the wheels off the Batmobile, Bruce offered him a home and purpose, but Jason's superhero career is cut short with his untimely death at the hands of the Joker.
Tim Drake followed. Using his detective skills, Tim found out the secret identity of Batman and argued the need for a new Robin. After the events of Battle for the Cowl, Drake reluctantly became Red Robin.
Stephanie Brown (aka Spoiler), briefly took on the Robin mantle in place of Drake. Black Mask captured and tortured her and she seemingly died from her injuries. She was revived as Batgirl when her death was found to be faked.
Damian Wayne, son of Batman and Talia al Ghul, became Robin after the events of Battle for the Cowl. Damian had learned to kill at a young age, having been trained by the League of Assassins. He was killed by the Heretic, his own artificially aged clone.