Many or most internet-enabled software companies have a vested interest in allowing their customers to bring data into or out of their own app. But to do this safely and efficiently, the company must enable specific pathways that are controlled and measured.
An API comprises the set of tools that handle access control ("authentication"), move data to/from specific database tables relating to the structure of the app (i.e. "People" vs "Companies" in a CRM), and track data movements precisely ("API logs").
We like to explain APIs with this metaphor: Think of your app as Manhattan, and the API as the bridges and tunnels that bring data to/from the other burrows.
If you'd rather learn about APIs visually, watch this excellent video from Mulesoft.
If you're a developer but not that familiar with APIs in a technical context, this video is really helpful.