Application Programming Interface (API) is indispensable to any web solution, software module, plugin, web application, or web design as it lays down the protocols / standards to access a web based app. Whether you want to create a web-based software application, or a web-tool, API plays a critical role in the success of your business endeavors, as it is an integral component that facilitates reliable interaction between computer hardware, databases, and software components.
Users of your software can only see the interface of your software. API is developed to seamlessly integrate different applications and work in the background so that the user is bothered about the flow of control between different applications. For example, users of your e-commerce website, who use a credit card for purchase, do not see the dialog between your software and the credit card's authentication program. This is handled by API's.
Importance of an API can be gauged from the fact that almost all powerful apps these days are powered by highly specialized APIs, each developed to execute a specific function.