The only apps that would continue to run in the background are apps that utilize the GPS (indicated by the location/GPS logo in the notification bar) or audio apps that are currently playing (indicated by the Play logo in the notification bar).
If you double click the home button, you will bring up the multitask bar, which will display recently used apps, which have been "frozen" in memory ready for you to jump back in where you left off. They are not actually running (unless they are one of the above mentioned apps), just taking up some room in memory.
To clear these apps from the multitask bar, hold the icon until it wiggles then hit the red minus button.