Android :: Phone Deactivated Or In Standby Mode?

When I deactivate my Android phone, I push the power button, and the screen then shows nothing more, i.e. it looks deactivated. But when I push back the power button, say, 2 hours after, it directly notices me that I have a new message, if so. How can it know this, if it was shut down? Is the whole device in some standby mode when you turn it off like that? So, does the antenna still works in this case? Is it the right way to use an Android phone? I.e. if somebody calls me, will I be noticed of this call if I deactivate my phone as mentioned above? Or should I let it always turned on for real (i.e. with screen acting, etc.)?