API as a service is a Software as a Service (SaaS) primarily exposed as an APIs (application programming interfaces). It allows users to explore functionality of Web Services such as Google Maps, Payroll processing, Credit Card processing, etc.
These API's normally provide multiple entry points for API calls ranging from REST, WebSockets, XML web services or TCP/IP.
Not to be confused with API hosting as a Service, which is a platform allowing API as a service services to expose their API endpoints as part of an API hosting platform.
References
API as a service Wikipedia(Text) CC BY-SA