Would I be using REST or would my app be considered RESTful if I was using the Google Volley library to connect to a database on localhost?
No.
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REST has to do with working with resources over the Internet; LocalHost doesn’t really qualify as such, unless perhaps you’re running a web server on LocalHost.
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Google Volley is a client-side library. It can potentially connect to RESTful resources, but it, in and of itself, does not make something RESTful.
If you simulated a RESTful resource on LocalHost, and then connected to that with Google Volley using RESTful protocols, then yes, you could probably call that RESTful.
Further Reading
How I Explained REST to My Wife