Well, first things first:

1. API--in the most generic sense stands for application program interface. It is used very liberally to describe the way a piece of software wants you to interface with it. It can be a set of functions or methods you can use to activate functionality in a 'black box' software product. Or it can be a toolset provided by a company to allow you to use some of their services. For example, Amazon has web services that have an API. You can use this write programs that leverage Amazon data.

Update: Link to definition in the webopedia and wikipedia.