Standard Definition:
A system access point or library function that has a well-defined syntax and is accessible from application programs or user code to provide well-defined functionality.
Simplified Analogy:
Simply speaking, an API is a set of rules and protocols that allow different software applications to communicate with each other. For instance, if a website wants its users to be able to make payments, instead of worrying about how to build their own payment portal, they could just contact an already established payment service provider and connect to their service via an API.
It can be compared to a business contract that specifies the terms of engagement between parties. Just like a traditional contract or agreement lays out the guidelines for interaction, defines responsibilities, and sets the expectations between businesses or individuals, an API does the same for software components, applications, or companies. APIs require careful design and adherence to ensure successful collaboration and can serve as critical evidence in to demonstrate compliance, intention, or the nature of relationship between two applications or companies.