In the main continuity, 5 individuals have assumed the role of Robin:
a young acrobat whose parents are murdered. He is taken in by Bruce Wayne and trained to fight crime as Robin. Dick grows up and drops the Robin identity and creates a new one, Nightwing, to escape living in Batman's shadow.
a street orphan caught stealing the tires off of the Batmobile. Batman offers him the now vacant role of Robin after he helps Batman catch some robbers. Jason Todd is killed by the Joker and then eventually replaced. Years later he returns as the Red Hood.
a boy with incredible detective skills who deduces the identities of Batman and Robin. Batman even acknowledges that someday Tim will surpass his own detective skills. He later shifts into the role of Red Robin.
Tim Drake's girlfriend who also was the Spoiler fills in for a short time after Drake resigns. This was retconned out for the New 52 continuity.
the son of Bruce Wayne and Talia al Ghul. After Batman's apparent death in Final Crisis, Dick Grayson takes on the role of Batman and places Damian as his Robin. Eventually, Bruce returns and Dick goes back to his Nightwing persona but Damian stays Robin.
If we are including elseworld stories, I wouldn't even know where to begin except to note Carrie Kelley from the Dark Knight Returns as the most prominent portrayal.