The turntable is driven by a motor under the bottom of the cooking cavity. A plastic coupling that slides over the motor shaft, and that's what turns the turntable. In some ovens the coupling is a separate small piece that the glass tray sits on. Other models use a large wheeled plastic ring that rotates the tray.
The usual reason the turntable stops is that the coupling breaks or wears out. The motor shaft is D-shaped, and sometimes the plastic coupling cracks so it doesn't grip properly. Or the plastic wears so that the opening gets rounded and won't grip the shaft. Then even though the motor is running, the coupler doesn't turn. You can usually lift the coupling off the shaft to check it. In some ovens the coupler can only be removed by taking out the turntable motor, but that's not common.
A second possibility is that the turntable motor has failed. In most ovens there is an access panel on the bottom that comes off to get at the motor. You can check the motor with an ohmmeter across its terminals to see if it's good. (Usually I advise against trying to repair a microwave yourself, but
replacing a turntable motor usually doesn't involve removing the cover.
It's not a dangerous repair.)
If the motor checks OK (not showing infinite or very high resistance with an ohmmeter), you can check for power at the motor when the oven is running. This is a little more difficult because it means there will be AC line voltage present, so caution is needed. If there isn't power, it means there's some electrical problem to be solved, like a bad connection or trouble on the control circuit board.
The coupler and motor are both replaceable, although not always easy to find. Sears parts
are U.S. sources for many brand parts. You can also search online or check with appliance servicers in your area. If the problem turns out to be electrical, you should leave the servicing to a professional. Internal repairs involve hazardous voltages and possible microwave leakage if done improperly.
Depending on what you find is wrong and the cost of the parts (and labor, if you aren't able to fix it yourself), it may be more economical to replace the oven. For a basic counter-top model, it may not cost much more just to buy new, and you'd have a warranty. In the meantime, you can still use the oven, but you'll need to stop the cooking and turn the food occasionally to get even cooking.
Good luck, happy cooking, and thanks for using Fixya! If this has been helpful, please take a moment to leave a thoughtful rating.