Simply put, 451 degrees Fahrenheit is the temperature at which paper burns. In this grim look at a fascist future, firemen have taken on a new duty of starting fires as well as putting them out (complete with flame throwers), specifically with the task of burning all books, as way of suppressing independent thought and action in the public.
The film's story centers around a young fireman, Guy Montag, who finds himself questioning his job, even as he encounters a beautiful young woman, and learns about an underground of rebels who each memorize the entire contents of a book, so that they can preserve it even without the use of paper...