I often read here at prince.org of a certain song by Prince being only a demo. What exactly is meant by that? I know what a demo is in and of itself but with Prince having access to a multimillion dollar recording facility where were these demos recorded? Done on a home 4-track in a bedroom or at the kitchen table? An 8-track? A rehearsal? What is the quality? Are they just basic tracks done at Paisley Park to flesh out song ideas?

Demo's.
Prince does not do them often.
He sumtimes tinkers on songs so much he ends up with different versions, or reworks them years later.

The song "Last Heart" is a demo he never got around to finishing.

mistermcgee said:

A demo, in Prince terms, is an early version of a song, or a rough mix done on-the-fly.

He might go into the studio just to get the idea out of his head and make space for the next song. But it might just be a rough cut and he might revise or totally revamp a song several times until he is happy with it.