Whenever I ask people about the difference between an API and a library, I get different opinions. Some give this kind of definition, saying that an API is a spec and a library
we have first to define an interface ...
Interface :is the means by which 2 "things" talk to each other and exchange information. "things" could be a (1) human or (2) a running code of any sort (e.g. library ,desktop application , OS , web service ... etc).
if a human want to talks to a program he need Graphical user interface (GUI) or command line interface (CLI). both are types of interfaces that humans (but not programs) would like to use.
if however a running code (of any sort) want to talk to another running code (of any sort) it doesn't need or want a GUI or CLI ,it rather need an Application Programming Interface (API).
so to answer the original poster question : library is a type of running code and the API is the means by which this running code talk to other running codes.