Here's the long and short of it. When you are running an app in iOS 4 and you press the Home button once to leave it, the app doesn't quit. Instead, it goes into suspended animation, like the scientists in cryogenic hibernation in "2001: A Space Odyssey." The app simply stops receiving events from the system; its run loop isn't looping. The app is both backgrounded and inert; but it is still running, in the sense that its resources and interface are still present, so that it doesn't have to be relaunched from scratch in order to resume. That way, when you come back to that app, no matter how, the app can simply pick up doing what it was doing when you left off, instantly.
This behavior is fast app switching, and is most of what Apple calls multitasking on iOS 4. (It is not the whole of iOS 4 multitasking, because some apps with specialized functionality will register with the system to be allowed to go into suspended animation with one thread still active; such functionality is strictly limited to playing audio, location detection, and voice-over-IP. But in this article I'm not concerned with those situations.)