there are 2 types of backgrounding in iOS.
first is multitasking, there are only certain things that can happen in the background- navigation, music, an VoIP.
and those have to be written so that only the things that are necessary are happening in the background, so your GPS app will still tell you where to go, but it's not updating the map display
second is fast app switching, that saves the state of your app when you close it, and stores it in memory. That way you can swap back and forth between a few programs quickly. As the device needs memory, it automatically dumps the oldest apps first. so yes it does use memory, but it's a "soft" use.
notifications are just fancy text messages, the majority of them don't come from your phone, they are generated somewhere else, and then sent through apples servers. so your phone could even be powered down, and when you turn it on you'll get them.