If you're telling a story, there comes a time when you have to explain to the audience what's going on. Exposition is unavoidable, but if your characters merely start telling the story to each other, the script will feel false. People just don't talk like that.
One technique frequently used to cover exposition and make the scene interesting is to use conflict. Every scene should have some form of conflict, but by placing conflict at the heart of the scene, the exposition can seem a natural resolution instead of a hokey intrusion. For example; rather than have the characters discussing the story directly and explaining it to each other, place them into conflict with the information, forcing the discussion into more dynamic areas. It is almost a form of redirection as the scene ends up about whatever the conflict is and not the exposition, which becomes incidental to the thrust of the scene.
In my mind, Aaron Sorkin is a television god and The West Wing a writer's masterclass. In the following opening scene from the first episode of Season Three, Sorkin needed to get across two pieces of vital plot exposition; One - that Josh will be looking after a group of students on a visit to the White House, and two - that the White House has suffered a series of security lockdowns over the last few days.
Watch how Sorkin gets this information across without boring us to tears.
Sorkin places Josh in direct conflict with the plot. The visiting students and the security crashes become obstacles to Josh by giving him the simple goal of going home at 5pm. Donna voices that obstacle, explaining why Josh has to stay by telling him, and us, what we need to know.
What is even better is that once Sorkin created this device, it was easy to feed that back and tie it to one of the obstacles. Why does Josh want to get out of the office quickly at 5pm? Because the security crashes make it hard for him to work in the office.
All this makes Josh's situation seem plausible and the need for explanation entirely justified, as well as a lot more interesting to watch. The scene is no longer about the exposition - it's about Josh trying to get out of the office.
Character goals and obstacles aren't only necessary when creating the central plot. They can be invaluable to give life to a single scene, allowing a change of focus and turning them into complete stories with a beginning (Josh tries to leave), middle (Donna uses exposition to make him stay) and end (the crash happens - forcing Josh into lockdown with the students). This scene does just that, ending with all the pieces in place for the main story to unfold.
In approaching scenes in this way, focus is taken off exposition in a beautifully entertaining piece of slight-of-hand.
