"Now, I want a nice fair game, all of you."
-- Madam Hooch, before a Slytherin vs. Gryffindor Quidditch match (PS11)
Madam Hooch is the Quidditch coach and flying instructor at Hogwarts. She referees the Quidditch matches between the House teams.
She is an expert on racing brooms and can be as enthusiastic as any of the older House team players are, and while admiring Harry's Firebolt, she mentioned that she first learned to fly on a Silver Arrow (PA13). This is the only hint we've had as to her age; the Silver Arrow was manufactured very early in the twentieth century before mass production was available (QA9).
Great knowledge of Quidditch and coaching.
*Madam Hooch's first name is given as "Rolanda" in the Trading Card Game and in a non-canon article on Pottermore (i.e. not one written by Rowling herself). The Internet Movie Database website listed her first name as "Xiomara" prior to the release of the first film, but removed it after the film's debut for unknown reasons. Obviously they can't both be correct. Neither is considered canon until independently confirmed by JKR.
In the Films:
Harry Potter and the Philosopher's Stone: Madam Hooch is played by Zoë Wanamaker.