An API is an Application Programming Interface, though as is the way with most acronyms, spelling them out makes us none the wiser.
The important bit of this acronym is the "Interface" part, because it describes what an API does. An API exposes part of a programme to another programme. It enables one programmes to ask questions of another programme and receive answers it can understand.
For example, Google Maps is a web site that you may be familiar with. When you enter an address, it will show you a map centred on that address. However, Google Maps also has an API which allows us to place maps on our own web sites. The question that we ask is the computer equivalent of "Can I have a map for these co-ordinates please?". The API returns the information required to build the map and draws it on the page of your web site.
You can use other people's API on your own web site or you could build and API to let other people access your own information and display it on their website.
For example. If you sell widgets on your you could build an API that exposes some of the information about your stock levels and prices to your business partners.