API is the word behind every headline like "the company opened its model to developers." It describes how two programs talk to each other with no human in between, through exactly defined request and response forms. Understand once how the counter works, and next time you'll see who's really standing behind someone else's app.
Picture sitting down at a restaurant table to order dinner. You don't walk into the kitchen to check what the cook is actually doing. You order from the menu, the waiter carries the order through the swinging doors and comes back with an answer: the finished dish, or word that something on the menu ran out. You never see what actually happened in the kitchen. And you don't need to.
That's exactly how two programs work together when they talk through an API (short for Application Programming Interface, literally an interface for programming applications). One sends a request in a precisely agreed format, say "give me tomorrow's weather forecast for Sofia." The other processes it in its own code, which you'll never see, and returns an answer in that same agreed format. Neither side knows, or needs to know, exactly what's happening on the other end.
When a company announces it's opening an API for its product or service, it's doing something specific: it stops cooking only for its own dining room and starts supplying any restaurant that wants to buy from it. Come in, order from the menu, we'll serve you. Until then the product was a dish served only at its own tables. From that point on, any other app can call it from the inside, invisible to the person actually using it.
I won't dress it up
When you read that some company is opening an API, don't think technology. Think of a big food supplier who decides to sell half-made dishes to every restaurant in town, instead of serving only its own tables. It no longer shows itself to the customer. It sells ingredients to others, who then serve you something completely different, without ever mentioning whose kitchen it came from. That's why so many "new" and "independent" apps actually fill their plates from the same kitchen out back. The storefronts are many, the kitchen is often one.