Just because the exposition is the beginning of the story doesn't mean that the exposition starts at the actual beginning of the story. Many longer works begin when (or right before) the conflict or problem begins. Other stories, especially shorter works, however, may start in the middle of the conflict, a technique called "in medias res." In this technique, the character may have been dealing with the conflict for years, but the reader is introduced to the story when things start to get exciting. Some stories may even begin at the ending and use flashbacks to tell the story. Whichever technique is used, the elements remain the same: the beginning must introduce the setting, characters and conflict.
The main purpose of exposition is to establish what the story is about. The introduction should include the main character. While character development happens throughout a story, the introduction will show some of that character's qualities. Often, the introduction establishes qualities of a character that change by the end of the story. That character will also be somehow involved in the conflict. While the beginning establishes a conflict, that conflict will deepen and eventually be resolved through the other parts of the story.
Exposition is just the first part to the plot structure. While there are variations on plot structure, the most frequently used model is Freytag's Pyramid. Once the exposition establishes the conflict and characters, the rising action further complicates the problem, with additional obstacles and twists. The rising action is followed by the climax, or the top of the pyramid, where the change starts to take place. After the climax, the conflict is on its way to resolution through falling action and then finally reaches a conclusion.
When writing, the beginning is essential to drawing readers in. Exposition must introduce the characters and conflict, but what else? How do you begin the beginning? Answering this question often means looking not just at how to begin, but where to begin. Starting in the middle of the action is often a good idea for shorter works that don't have the space to describe a conflict from the beginning. Longer stories will often start right as the conflict begins. No matter what the length of the story is, the best stories often start where the main character becomes involved in the conflict. For example, science fiction stories don't start with how the technology was developed, they start with the main character interacting with that technology.