...if an app is running in the background or not.
First way: Get the $0.99 iStat app for iPhone from the App Store and sync it to your device. Start iStat and make note of the processes running. Then, go start a few more apps and open iStat again (iStat does NOT multitask yet, BTW). If the apps you opened support multitasking, then they will be in the list of processes.
Second way: I've noticed that when you double tap the home button for the app switching display, apps that actually multitask will sort of 'turn' away from view much like Fast User Switching in OS X.
[edit] oops, I don't think that really means anything, after further experimentation. 
The screenshot shows Photos (MobileSlideShow), Preferences (Settings), and iBooks (of course, iStat is running currently) all running at the same time.
First way: Get the $0.99 iStat app for iPhone from the App Store and sync it to your device. Start iStat and make note of the processes running. Then, go start a few more apps and open iStat again (iStat does NOT multitask yet, BTW). If the apps you opened support multitasking, then they will be in the list of processes.
Second way: I've noticed that when you double tap the home button for the app switching display, apps that actually multitask will sort of 'turn' away from view much like Fast User Switching in OS X.
The screenshot shows Photos (MobileSlideShow), Preferences (Settings), and iBooks (of course, iStat is running currently) all running at the same time.