The api program shut down in 2006... it was a key which allowed to you access google searches programatically... I'm not sure if it offered you anything extra that you couldn't get simply by querying Google... googlerankings.com is an interesting website which requires the use of a Google api key...
There are several APIs available. Are you talking about in general? Here is the definition... API is a abbreviation of Application Program Interface. A set of routines that an application uses to request and carry out lower-level services performed by a computer's operating system. Also, a set of calling conventions in programming that define how a service is invoked through the application. I hope that's helpful enough to understand what does API stands for.