What could cause a chainsaw chain to stop moving?
My saw has a brake that can protect me in case of a kickback. It sits in front of my left hand. Push forward to engage, pull back to release.

Or it could be a maintenance issue:


Do you fill the bar oil when you fill the gas? 
Do you clean the machine after using it? Including the drive gear and the slot in the edge of the bar?
Do you loosen the chain when putting it away? As the chain cools it contracts, and you don't want to let it get overtight. I store it loose; snug it up before using; adjust after 5-10 minutes use, when it's warm; the release when putting away.
If you remove the bar &amp; chain, does the drive gear spin?