An exposition takes place when you "expose" something — like at a fair or convention for showing off goods, art, or similar wares. An exposition can also be the section of a story that explains the basics of the tale.
In classical music, the first part of a piece is often an exposition, in which the composer lets you hear the tunes and themes that will get worked over during the rest of the piece. Most movies spend a lot of time at the beginning on exposition, as well. The director wants you to know where the story is set, who the characters are, and what it is they want. Exposition is important to set the scene, so the rest of the story makes sense. Exposition can also mean a big fair, often called an "expo." ComicCon is a famous exposition.