An API is really no more than a set of method signatures and other information necessary to use a class or set of classes. The API is totally distinct from the implementation. For example, a number of vendors have implemented the servlet API, each in a different way. The API by itself has no implementation.
Library (being JAR Java library) is a unit of deployment of one, many or part of API.
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A library is a collection of pre-written code that provides reusable functions and routines for developers to use in their own programs. An API, or Application Programming Interface, defines the methods and protocols for how software components should interact with each other. Essentially, a library is a set of code while an API is a set of rules for how that code can be utilized.
The difference is in the punctuation.
Library's is possessive. For example, you would use this form to say, "There was a cat sitting inside the library'sbook-shoot."
Libraries is plural. For example, you would use this form to say, "There are seven libraries in the city."
Libraries' is a plural possessive. That is, you are saying that something belongs to several libraries. For example, you would use this form to say, "The libraries' total book collection is large."