An API (Application Programming Interface) allows other websites and applications to work directly with Simplelists without the need for manual intervention. For example, you might run a separate contact database with all your group members in. The API allows that database to update Simplelists without requiring list manager involvement.
The easiest way to use the API is with a module that has already been created for an application, if it exists. Currently the only such available module is for CiviCRM, the details of which can be found on the CiviCRM API page. Alternatively, you can create your own by interfacing directly with the API interface. Be warned: this option is for programmers only!