The garage door Drive Chain Sprocket Won't Move
Inside the garage door opener there is going to be a gear reduction system to adapt the motor speed to the chain sprocket speed. This speed reduction system also increases the motor torque at the sprocket so that a relatively small motor can easily pull the chain and move the garage door up and down. 

Trying to turn the sprocket directly can be difficult if the gearing system uses all normal straight gears. On the other hand if the gearing system uses a worm gear drive it would be next to impossible to directly turn the sprocket. This picture will explain a worm gear drive. 



So you should not jump to conclusions at to if the opener is broken or not. It will require testing. Sometimes the best way to do that is to unhang the opener and chain bar and test the assembly on the garage floor.