API stands for Application Programming Interface and is essentially a way for two systems to communicate and either push or pull data from each other in order to achieve a task.
Some examples of Applications with API's are:
An API is a bit like a waiter in a restaurant. The customer (client system) needs to convey information to and from the kitchen (back-end system) in order to receive food, normally this would be a difficult task to achieve for the customer, especially with lots of other customers wanting to send and receive information as well. However, with a waiter (API), information from multiple customers can be transferred to and from the kitchen with a lot more ease as the customer does not have to get out of their seat, find the correct person to talk to etc.
So, in conclusion, an API makes the transferring of data/information between two systems possible or much easier through the use of agreed protocols and libraries that both systems can understand.