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I notice the existence of something called API, but do not have a clue what it is.

Could someone please explain in detail what it is.

Why would paypal require this as part of their website payment services?

API stands for Application Programming Interface.  It's basically a way for you to systematically communicate with another computer system.

Let's use our API as an example, if you had a list of domains you wanted to register, you could enter that list into a program you have on your desktop.  Your computer would communicate with our API computer, checking and registering the domains that are available.