What API means
API stands for Application Programming Interface. MDN's documentation describes it as a set of features and rules inside a program that enable interaction with it through software, as opposed to a human user interface. It also compares an API to a contract: whoever uses it knows what they can ask for, in which format and what they will get back.
On the web, most APIs run over HTTP. A program sends a request to a specific address, called an endpoint, with the parameters it wants, and receives a response, almost always in JSON. To know who is asking, the API usually requires a key or a sign-in via OAuth, and it limits how many requests can be made in a given period.
It is worth separating the API from the tool that uses it. Search Console has a web interface for people and, on top of that, an API for programs. The data behind them is the same, but access rules, limits and what can be automated differ.