Close app as opposed to letting it run in the background I'm not sure whether this is a Galaxy Nexus question or an ICS question, but here goes: 

Suppose I'm using some app and then I move to the home screen, rather than shutting down, the app is left to run in the background. Is there a way of exiting an app such that it gets closed upon leaving it rather than being left to run in the background?
This is really a question about Android in general, and is based on a misconception of how Android handles background apps. Check out this XDA post for a nice explanation of why processes sitting in memory in the background really aren't a bad thing. You should only worry about exiting/force-quitting/killing an app if things start misbehaving.