In Fable, Mr. Gout only has a small role in the quest Collect the Hero Dolls. If the player enters Bowerstone South during the day and has obtained at least one hero doll, a small child will approach and advise the Hero to take any dolls he owns to the teacher at the school, who owns an exceptionally rare doll. Once the player has obtained all six hero dolls (seven in TLC), he can trade them to Mr. Gout, who will surrender his rare Jack of Blades doll.
In Fable: The Lost Chapters, the Hero will encounter Mr. Gout again, who will explain that Bowerstone's mayor, Lady Grey, sold many of the books in the school's library in order to fund a building project. He will ask the Hero to bring him any spare books in order to replenish the school's library, starting the quest Book Collection. After the Hero donates a book, Mr. Gout will either teach a lesson or perform a dramatic reading of the text (or the students will act it out). As a reward, Mr. Gout and his students will give the Hero a Bright Wizard Hat or a Dark Wizard Hat, depending on the books donated.
His name is most likely a reference to the Gout disease.
Mr. Gout is chronically late in the mornings and will always arrive at the school well after his students.
You can get Mr. Gout to follow you both in Fable and Fable: The Lost Chapters and he will do so despite how evil you are.
During Book Collection, if the "Book of Spells" is donated, Mr. Gout will remark, "The book of spells? This isn't some potty school of wizards, you know!" This is likely a reference to the Harry Potter series of books.