An API is the set of addresses and rules a system publishes so that another system can talk to it. In the context of payments, it is what lets your membership area ask the platform whether a given sale was approved, or lets your CRM create a charge without anyone opening the dashboard.

Communication is usually over HTTP with data in JSON, authenticated by a key that identifies the account. Unlike a webhook, where the platform notifies you when something happens, with an API you are the one asking.