Stands for Application Programming Interface. It enables communication between computer programs. GetResponse defines it as An Application Program Interface (API) used by one program to manage another by remote control. For instance, with an API key you can access GetResponse Web services and manage contacts individually or in bulk operations and provide access to summary campaign results.
If you are a developer, and you need to develop an application that will be able to communicate with your email marketing software, make sure that it provides an API.