Exposition of a story is the part in which characters (and their backgrounds) are introduced, and the stage upon which the action will play out is set.
In the case of Stephenie Meyer's Twilight, this involves detailing Bella Swan's move from Phoenix to Forks and the reasons behind it. In order for Bella to meet Edward, she must start attending classes at the local high school, which is another important part of the exposition.
The exposition of this story also involves the introduction of the Cullen siblings, who are widely perceived to be strange, introverted, and unapproachable.
I would argue that the final stage of the exposition of this story would be Bella and Edward meeting, his initial extreme discomfort in her presence and her ultimate realization that he is a vampire. The stage is now set for the love story and adventures that will fill the rest of this book.