"Eveline," one of the stories in James Joyce's Dubliners, follows a young woman named (rather predictably) Eveline. Before discussing the exposition of the story, however, it's important to understand what exposition is. Exposition is generally background information and explanation given to the reader early on in a narrative; exposition "sets the scene," as it were, providing necessary context about the world and story that the reader is entering. As such, exposition provides the scaffolding for the main narrative.
In "Eveline," the exposition is the information we get about Eveline's background. As it turns out, that background is a pretty grim one. Joyce tells us that Eveline's mother and brother are dead, and her father is an alcoholic with tendencies toward violence. We also learn that Eveline has a dead-end job at a store, makes a meager wage that she must forfeit to her father, and also must take on the responsibilities of her dead mother and care for her two younger siblings. Finally, we learn that Eveline has recently become engaged to a young man named Frank who pledges to sweep Eveline off her feet and take her to Buenos Aires.
Through this exposition, we learn two major plot points: first, Eveline's present life is miserable and devoid of promise. Second, Frank represents Eveline's only means of escaping her present prison. With this knowledge in mind, we're ready to digest the main thrust of the narrative.
A few more bits of information about exposition: It is usually in the first few paragraphs of a story, providing the necessary background information.
It sets the scene, establishes the situation, and dates the action. It usually introduces the characters and the conflict, or at least the potential for conflict.
Note how the opening 3 paragraphs of "Eveline" does all of this. You can make a list of what exposition does, find that information in those paragraphs, and then write it down next to each item.