The full form of `API` is the `Application programming interface`. What it means is that an interface for an "application" to another "application". Like User interface(UI) means an interface for a user(human) to an application, API is an interface for applications.
Now let's talk about "programming" word in the definition. It means that this interface can be used for programming. This interface can be used in a programmatic i.e automated way by another application. Thus an API can be used by an "application" to "communicate" with another application in a scalable way.
A simple example of usage of API can be seen in the air ticket booking sites where when you search for a ticket it shows you results in different airlines. Different airlines maintain their own data and the data aggregators(ticket book company like Yatra.com) calls an API that aggregates all this information and provides to the aggregators. In a way, data is coming from a 3rd party source and being displayed by the aggregator. This is done via APIs.
APIs are extended by many businesses who either want to offer their data for a fees or some service. An example can be govt institutions offer multiple APIs which can be consumed by businesses for their use. Payment gateways offer their payment service in the form of APIs. You can use the APIs to make a payment, see the status of a transaction or your account balance.
In the finance industry APIs are getting more and more popular as APIs act under the surface thus can lead to smooth customer experience.
Please watch the below video for a more detailed explanation.