API is an acronym for Application Programming Interface. An API is a way for two applications to talk to one another. I like to think of an API as a data pipe that connects two systems together.
Probably most software and hardware you are using uses an API because the server has to talk to the application somewhere along the line. For the most part, most systems use an API to do a lot of the data processing and keeping data secure before it is inserted directly into the database.
Most companies are now building an API for at least mobile applications or other devices to call. A well designed system will use an API that can do load balancing and data caching to more efficiently use resources and better deliver a consistent user experience.