Application programming interface (API) is basically a how-to guide for building software applications. Some APIs are guidelines of general best practices, while some are very specific, detailing how to implement a single feature into an app.
Let's take Touch ID, for example. Originally, Apple introduced it on the iPhone and you could only use it to unlock your phone or make purchases in iTunes. Then, Apple opened up the API and made it available to developers. Now, you can use Touch ID in a variety of third party apps, such as banking applications, where you want to keep what's inside extra secure.
At developer conferences, companies introduce a lot of new APIs. For example, at this year's Google I/O, Google introduced new APIs for Android Pay (its mobile payment platform), Android Auto (so apps can integrate with new in-car infotainment systems), and an Awareness API (which lets apps access sensor data from your phone to better assist you in whatever task you're trying to do).
As Apple's developer conference approaches, you'll likely hear more talk about APIs.