An API is a service that allows developers to connect and communicate by the use of a protocol. The protocol allows the connection of both integrated and separate software, the interface being the medium by which the two systems are able to communicate.
APIs are vital for communication and you will find them all over the place such as Twitter API, Flickr API, and Maps API. To take an easy to understand example of how vitally important they are, imagine if you could not copy and paste from notepad into Google Chrome and you had to re-enter the text by hand; an API makes this happen without wasting time on rewriting your text.
Most operating environments provide APIs as do most large websites these days. These APIs are provided to make it easier for developers to link their own software into the operating system or website. For example, the Amazon API is used by developers to create specialized online store-fronts for ecommerce purposes.