The google API lets software developers and web dev programmers to write applications that can interact with google. For example, someone could write a PHP script that lets you input a URL and it shows the google PR of that URL. In other words, your program queries google behind the scenes for this info, and google provides it without revealing any secrets as to how PR is calculated.
However, you need what's called a google api key to do this. A key uniquely identifies you as a programmer. You use this key in the software that you write. It is a way for google to keep tabs on what software is querying it.
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