Roberta Sparrow is an important character in Donnie Darko.
Little is known of her background. However, it is said in the movie that she was once affiliated with a church or religious organization of sorts, but left said organization sometime before becoming a science teacher at Donnie's school. By the time the events of the film took place, she was a seemingly senile, elderly woman, whom wandered around Middlesex and seldom seemed aware of her surroundings.
Despite at first seeming an irrelevant, background character, as the film progresses, she proves to be an important aspect of what unfolds. She seemed to have a full understanding of what was occuring, more so than anyone else, who all seemed oblivious to the situation, except for Donnie himself.
The most important aspect of Roberta Sparrow is the book which she wrote, The Philosophy of Time Travel. This fictional book, and the few pages of it available, is essentially the only canonical explanation for what happens in the film. The book itself is given to Donnie by one of his more amiable teachers, and serves as a substantial guide for his journey.
Roberta Sparrow is thought to have been in the same situation as Donnie at some point in time.
She is thought to have died shortly after the events of the movie.