Diagnosis
The drive motor underneath the floor of the cavity has likely failed or become obstructed. Yeah, this one's common.
How to Fix It
First, remove the glass tray and the plastic roller guide ring underneath it. Check for any food debris, melted plastic, or anything jamming the mechanism. Manually rotate the little coupler (the white plastic nub) in the center of the floor. If it turns freely, the motor might be dead. If it's stuck, clear the obstruction. You can also try running the microwave empty for a few seconds and watching/listening to see if the coupler attempts to turn.
If That Didn't Work
The turntable motor is probably burnt out. It's a small, geared motor. You can access it by removing the outer cabinet and the waveguide cover inside. It's held by a few screws and has a simple plug connection.
Heads Up
Running the microwave frequently with a non-spinning tray can cause hot spots that damage the waveguide cover (the mica plate on the side wall). Check that cover for any signs of burning or arcing and replace it if it's damaged.