API stands for Application Programming Interface. You can think of it as a common language that lets different software systems communicate smoothly, without needing to know how the other is built. More technically, an API is a set of rules and protocols that determine how one application can request data from another.
When running on a server, an API is a set of coded routines that receives requests from and sends responses to other programs. API designers implement that code through standardized programming statements that expose functions that make sense for accessing the platform in question.