API stands for Application Programming Interface. Think of this as a contract, or an agreement, between one piece of software and any other piece of software that wants to make use of it. APIs are all around you. Most every well-known software service (think of big names like Microsoft and Google online apps) have APIs you can use. This allows you to extend and combine pieces of software. Coding by jigsaw.
There are many types of APIs. Library files have been around since the early days of programming, and the ways of calling code at one location from code at another location grew over time. One very popular way of doing this today is to use HTTP (web-page-like) requests to signal between applications. This is (very loosely) known as RESTful web services.