An API is a set of agreed ways for one piece of software to ask another piece for help. It is like a restaurant menu: it shows what you can request and how to ask for it.
A weather app may ask a weather service for today's forecast through an API. The service returns information in a format the app knows how to display.
An API does not mean a person can access everything inside a service. It defines particular requests and responses, and the service can require permission before it answers.