API stands for Application Programming Interface, i.e. API is the way for an application to interact with certain system/application/library/etc.
For example, there are API's for OS (WinAPI), API's for other applications (like databases) and for specific libraries (for example, image processing), etc.
APIs are usually developed in a form consumable by a client application. For C/C++ applications, it a set header files and dynamic/static libraries. For Java - set of jars. And so on.
