In the world of modern business, we often here the term, "Elevator Pitch," which is essentially means, how to explain your topic to someone new in that 30 to 60 seconds that you might have that person as a captive audience on an elevator ride. That is provided, of course, that the "audience" doesn't jump off at the next floor to get away from you!
So, what's the API Elevator Pitch? In my mind, its this: An API is a piece of code that allows one software application to talk to another software application. Think "Facebook" sharing. The structure is already created for you to share posts to whatever Facebook accounts you manage. Have you ever noticed Twitter posts on pages you visit? Same thing; an API is used to move that data from the original Twitter Post to a site you watch. I watch the eBay for Business Blog. It has a Twitter feed on the right side navigation.