Question: 1. What is API?
2. What is Importance of it?
3. give me example of it?
4. how can developer use API?

Answer: 
For a broad answer to your question you can check wikipedia: 

<http://en.wikipedia.org/wiki/Application_programming_interface>

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> An API is an abstraction that describes an interface for the interaction with a set of functions used by components of a software system. The software providing the functions described by an API is said to be an implementation of the API.
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An application programming interface (API) is an interface implemented by a software program which enables it to interact with other software. It facilitates interaction between different software programs similar to the way the user interface facilitates interaction between humans and computers. An API is implemented by applications, libraries, and operating systems to determine their vocabularies and calling conventions, and is used to access their services. It may include specifications for routines, data structures, object classes, and protocols used to communicate between the consumer and the implementer of the API

What Good Are They?

APIs allow us to tie functionality into websites that would otherwise take an enormous amount of time and effort to build from scratch. APIs are a sanctioned way of reusing someone else’s functionality without having to build it all yourself.

<http://en.wikipedia.org/wiki/Application_programming_interface>