The Android model is perhaps the most interesting, a hybrid system that's fairly open about allowing stuff running in the background, but at the same time aims to be completely invisible to you, the end user, so that you don't have to actively manage whether an app is open or closed. It's a little complicated so we talked to an Android engineer about what's going on inside.
What happens when I switch to another app?
The app you switched from doesn't stop—the process stays open. At least, as long as it can. When Android detects it's running low on memory, then it kills processes to free up resources. It asks apps to save their state when you switch, so that any can be killed at any point, then reopen like it was never killed when you return to it.
What can apps do in the background?
Android has two basic facilities for third-party multitasking—broadcast receivers and services. With a broadcast receiver, an app that goes into the background is basically asking to be told about an event, like if you move 500 meters, or your battery level hits a certain percentage. (That's how Google's apps that use push, like Gmail work—instead of pinging the server for mail constantly, it waits to receive a notification that mail has arrived.) That way the app can go away, and not use resources, but restart when something happens that it needs to act on.
Services are what you'd consider more traditional background processes, with apps running to play music, or do turn-by-turn navigation—they're essentially a request from the app to say it needs to run for X amount of time.
What can't apps do in the background?
Remember Android's garbage battery life when it first came out? That's because there really were no restrictions on what kind of resources an app could consume in the background. Since Android 1.5, apps running in the background are capped at using 5-10 percent of the CPU altogether—which is the only major restriction placed on apps in the background. The other is that it's not easy for app to push itself to the foreground—they're supposed to the window shade notifications system.
Hopefully... just hopefully. EVERYONE on this forum, ALL 50k+ users read this article and realize there is NO NEED FOR TASK KILLERS. Seriously mods. This should be made a Sticky and re-titled to "Be all and End all about Task Killers for Android"