An application programming interface, which is a set of definitions of the ways one piece of computer software communicates with another. It is a method of achieving abstraction, usually (but not necessarily) between higher-level and lower-level software.

Source: USOD

A way computer programmes talk to one another. Can be understood in terms of how a programmer sends instructions between programmes.  For data, this is usually a way provided by the data publisher for programs or apps to read data directly over the web. The app sends the API a query asking for the specific data it needs, e.g. the time of the next bus leaving a particular stop. This allows the app to use the data without downloading the whole dataset, saving bandwidth and ensuring that the data used is the most up-to-date available.

Source: ODH.