What is the purpose of a window function?
I've read that a window function is a mathematical function that is zero-valued outside of some chosen interval.
Can someone please explain what this really means?

It means exactly what it says, but I assume you really want to know why do we care?
The purpose of window functions in signal processing is to remove (hopefully) extraneous information, so that subsequent analysis produces more meaningful results. Checkout wikipedia (and other sources) for specifics, but windows functions do pre-processioning in the time domain to help improve results of later processing in the frequency domain.
This hopefully helps explain why there are so many different window functions.  Basically, different window functions are designed to help with different problems present in the time domain data.  Often there is no right/wrong window functions, there are just more/less useful functions for solving the problem at hand.