Well, in addition to all the answers, I am just adding an example.
As others said API stands for Application Programming Interface
through which softwares can interact with each other
. Note, not a human interaction.
Where it is used
An example: You are buying an item online through your credit card. You will provide credit card details and press 'continue' button. It will tell you whether your information is correct or not. To provide these results, there are lot of things in the background.
The application will send your credit card details to a remote application which will validate your information and send the result back to your application. API is used in this scenario.
I hope it helps for the beginners who don't understand really what API is.
ANOTHER EXAMPLE
Weather application
Without API - Weather application must open weather.com site and read the details as a human does.
With API - Weather application will send a message to weather.com and receive the result and then display it.
SOURCE - Various online resources