An API allows you to programmatically read or write information to one or more services. FileWave has a rich set of APIs to allow customers to build connections. There are also some vendors who produce integrations that use the FileWave API without you needing to work with the API directly.
Any time you want to extend what FileWave is capable of or integrate FileWave with other systems you may use APIs.
The articles linked to this article are a good starting point. Professional Services can also help with learning and using APIs.