The A/C compressor has a clutch mechanism that allows the compressor to be turned on and off. It is therefore normal to see the belt turning without the compressor running.
When the A/C computer wants the compressor to run, it sends an electrical signal to the compressor which operates the clutch. You should be able to hear a click as the clutch is operated.
If you watch this video @ 1 minute in, for example, you will see the disengaged clutch stationary while the pulley is turning.