An API is a defined way for two programs to talk to each other. One program sends a request in an agreed format, and the other sends back an answer or carries out the action. Think of a waiter between the kitchen and the table: you do not need to know how the kitchen works, only how to place an order.
In a business, APIs are what let your accounting package, your online shop and your customer system share information without anyone retyping it. When a new order arrives in the shop, an API can pass it to your finance system as an invoice. Most modern business software offers one, though what it allows, and how often you may use it, varies from product to product.