Application programming interfaces (APIs) provide the means for one software program to access the services of another.
Analogies abound for explaining how APIs work, but one that is particularly compelling is the wall socket analogy. Rather than having to hardwire a new appliance to the power source, a person need only plug it in to draw electricity. That person doesn't need to know anything about the concept of electricity or even how it's delivery — just that the plug fits correctly into the socket.
The electricity drawn through the socket represents the information contained in a piece of software; the socket, the API enabling a request to be made and answered; the appliance, the other piece of software developed to use the requested information in a novel way.