For the most part apps DO NOT run in the background. There are some exceptions for those that stream audio/video and GPS applications. GPS applications alert you to running in the background by having a little arrow in the top right corner of your "status" bar ( near the battery icon ). Audio & video apps you will hear them running.
To close an app press home, once. This will take you to the spring board. Tap the home button twice again to bring up your "recent apps" launcher. Tap and hold on an app, it will start to jiggle and a red X will be on the icon. Tap that to close the app.
Note: This is NOT necessary for most applications.