An Application Programming Interface (API) is a set of rules and specifications that allow two applications to communicate with each other. An API allows one application to make use of the features provided by another application, often without having to reinvent the wheel.
Though the concept of APIs seems to pre-date the term itself, the term was first used in a clear way in the paper Data Structures and Techniques for Remote Computer Graphics in 1968.
Today we rely on APIs every day—even if we don't realize it! A good example is Google Maps: when you search for directions from point A to point B using Google Maps' website or app, it uses an API provided by Google's mapping service provider Waze in order to find those directions using your location data (which could also be stored locally).
APIs matter because they allow different software programs and systems to interact with each other without having to re-create each other's functionality, thus saving time and boosting efficiency.