In short, I am writing an Android application that I want to have pull data from a remote database. I was looking into .NET web services, but this How to call a SOAP web service on Android question pointed me away from that direction.
Is REST as simple as writing some short PHP to handle something like https://www.example.com/data.php?employee=michael and have it return relevant XML or JSON data? That to me seems like what has been described to me by various sites and videos. Also, is this the best way to do this kind of operation over the internet for mobile devices, desktop applications, etc?
What you describe is pretty much what REST is about. Perhaps, however the URL to michael employee could be something like that:
https://www.example.com/employees/michael
Anyway, the idea with REST is that resources are exposed through URLs and that everything is stateless. This is a very good model to use in order to pull data from a database. It can be used both from mobile as well as desktop applications.