In the iOS simulator, you can "press the home button" with the CMD+SHIFT+H shortcut.
You can override the following methods in your UIApplicationDelegate implementation:
applicationWillResignActive: (Called when leaving the foreground state.)
applicationWillEnterForeground: (Called when transitioning out of the background state.)
Or you can use NSNotificationCenter and the equivilent notifications:
UIApplicationWillEnterForegroundNotification
UIApplicationWillResignActiveNotification
When iOS puts your app into the background, it is essentially paused, in that it is no longer running (with some exceptions for things like music players, GPS consumers, etc). But your render loop isn't doing anything. However, your app may get killed while in this state, so you probably want to do some bookkeeping to keep your apps state consistent. It wouldn't hurt to pause your app, save the users progress, etc, and then resume/restart and load that info when you come back.