API's are Application Programming Interfaces, they allow a system administrator to add additional programming code to another application, thus extending the functionality of the original application.  The API' also allow the applications involved to exchange information with each other to improve the quality of the application. 

Google Apps is Google's software-as-a-service offering for business messaging, collaboration, and security. Businesses of all sizes are using Google Apps to keep their employees connected, to improve their productivity, and to reduce IT costs. The Google Apps APIs enable developers to extend the functionality of Google Apps, integrate with other systems, or build new applications for their company and other businesses.