The language tools available for developers to communicate with and control software from code. For example, the API of a CMS usually allows a developer to connect to the CMS from their own code, and manipulate content objects.
A key part of the API is the event model offered to developers in which to "hook into" events happening in the inner workings of the CMS. Most systems will "raise" or "fire" events (sometimes called actions or triggers) in response to internal events in the code. Developers can configure the CMS to execute their own code (called event handlers) at these moments.
An API is part of the broader extensibility model of a CMS.