How does the Foucault pendulum work exactly, and would it work at all, if the Earth didn't rotate?
Well, for the basics of the Foucault pendulum, this wikipedia page does an adequate job describing how it works (Specifically read up on the Precession as a form of parallel transport section).
A Foucault pendulum is just like any other pendulum, nothing more than a weight attached to a wire; but to work well it needs to have a very long wire, with a really heavy weight. They're often attached to the ceilings inside very tall buildings, such as museums and cathedrals, so that they can hang a great distance and swing impressively slowly. These sorts of locations also tend to have fairly constant temperatures, which avoids expansion and contraction of the pendulum which would cause complicated variations in its period.
What makes a Foucault pendulum different from a normal pendulum is that it is attached at the top to a universal joint which allows the pendulum to rotate freely around its fixing point as it swings. Once you set one in motion, its direction of swing will rotate at a rate of about 0.2 degrees every minute. But in fact, it isn't really the pendulum that's rotating: the pendulum is swinging back and forth in exactly the same direction. It's the Earth which is rotating underneath the pendulum, which makes it appear that the pendulum is in fact changing direction.
Now, since I can't even conceive of a way to make a planet (or any other large enough body to build any type of pendulum on) that would not rotate, this is a purely theoretical exercise. Bottom line:
The pendulum would swing back and forth, and never deviate from its original arc (assuming no wind or other disturbances) if it were set up on a non-rotating body.
I keep thinking the pivot would rotate with the earth and therefore the Foucualt Pendulum would not work. But upon further thought the pendulum must track in same line. The pendulum stays constant because gravity keeps it lined up with the center of the earth while the earth's crust rotates the center does little and holds the pendulum on plane while the earth rotates under it. So if the earth did not rotate then the Foucault Pendulum would track the same line since the earth would not be moving under it. What else but gravity could keep the Pendulum course constant?