When your email tool shows data from your CRM, or your website sends an order to your payment provider, those systems are talking to each other through an API. It is the agreed way one program can ask another for information or actions, without a human clicking anything.
In the AI world, the word comes up constantly because every AI company offers its models two ways. There is the chat app you use in the browser, and there is the API, which is the same model made available for developers to build into their own products. When a tool says it is "powered by Claude" or "built on GPT", it means the tool talks to that model through its API.
Why you keep hearing it
Because APIs are how everything connects to everything. "It has an API" means other software can plug into it. "API pricing" is what companies pay to use an AI model inside their own product, usually priced per token. "API access" means using the model as a building block instead of as a chat window.
What it means for you
You will probably never touch an API yourself, and you do not need to. But when a tool you are evaluating "integrates via API" with your other tools, that is the real, automated kind of integration, not a copy-paste workaround.