The problem could be with your starter. It does spin when you hit the ignition but it fails to engage the flywheel fully so that car does not start. The best thing to do in this case would be to replace the starter. I would also advice that you consult a professional mechanic so that he can check out your vehicle and identify any other problem it may have that is causing it not to start.
Try disconnecting te battery, use a wire brush or sand paper, Brillo pad type thing to clean battery terminals and cable ends, reconnect battery hope for best. My cobalt does similar things sometimes, cleaning battery always fixes it