API stands for application programming interface.  API enables a software application which enables it to interact with a different software. It tells the other software what you want to do and then will  return the response of  back to you.

An example of API use,  most of us book flight tickets online and there are many websites through which you can search and compare the prices of different airlines, you just need to enter your date and destination and the website will list out all the flights. This is done using APIs provided by each airline. 

A trading API,  allows a trader to interact with a trading system. It allows you to execute directly on an exchange. This is particularly useful for traders who run algorithmic models on their own trading systems and want to receive live pricing and be able to execute trades.