Robin is the son of Lysa and Jon Arryn, he therefore inherited the title of Lord of the Vale when his father died. He is a weak and sickly little boy, and often has violent seizures.
Robin lives atop the Eyrie with his mother, shielded from the world. His overprotective mother has never let him go from her side, and he is quite literally still stuck to her teet. When Tyrion Lannister arrives at the Eyrie as Catelyn's prisoner, Robin is very excited at the prospect of seeing him "fly" out the Moon Door, a large hole in the middle of the throne room, with a drop of several hundred feet.
Trivia The character's name is Robert in the books, they changed it to Robin in the TV series to avoid confusion with King Robert.