Google's second link for API is 'Application Programming Interface'...
It's a set of interfaces that programs can use to get at the functionality of existing code. A game engine will have (at least) an API, but so will a lot of other things. You will use many APIs to make any non-trivial application (and a few APIs for trivial applications).
Yeah, think of APIs as being like a library with objects and classes (i.e. chunks of code, in layman's terms) that can be used by the game client (your main program) to allow you to do things not doable "top-down" by your high-level programming language (unless you design the API yourself, "bottom up"). High-level languages is what you call Java, C, C++, C# and so forth.
Library is basically when your code is in charge, your code uses the library, framework is when the framework is in charge and it uses your code and API is an interface for other programs to interact with your program without having direct access.
Ruby on Rails,ASP.NET MVC, jQueryand everything else web.. now also trying my hand on games.