An API is an application programming interface. This basically means that the application in question has provided an interface for programmers to use to access certain functionality programatically.
This, could mean lots of different things, but in regards to a web application, it generally means you make a GET request for certain data, and the application sends this data back. Of course, there can be more to it, for instance Google provides an API for Gmail that allows you to add new users (create new email addresses) to your (Google applications) domain.