API stands for Application Programming Interface. This is an interface that essentially allows two different systems to communicate with each other. It is akin to having two individuals that speak different languages with an interpreter in between. More specifically, the API takes the inputs or requests made by one system and applies it to the target system in a way that returns a useful response or action.
In terms of a trading API, it interacts with a trading platform to efficiently execute specified operations. High-frequency traders, as the name implies, use powerful algorithms or systems to make incredibly large numbers of transactions within fractions of a second. Because it would be impossible for a human to make that many operations manually, individuals or organizations instead leverage their algorithmic programs with trading APIs to achieve their desired results.