How can I fix a garage door motor that runs but doesn't move the chain?
It sounds like the chain might not be on the sprocket that drives it.  Get up on a ladder and look at the top of the unit. You should be able to easily see if the chain is off the sprocket.  Try running the opener while you are up there too.

If the chain has come off you will need to loosen the chain tension, place the chain back over the sprocket and tighten it to the correct tension. Your user manual should have details on how to perform this but usually there is a screw you adjust on the trolley that controls the tension. Make sure to disconnect your door from the trolley before you do this.

Here's  YouTube video that shows the process.