In iOS 6 and previous versions you have to double tap the home button twice to show the tray containing all the currently running apps in the background, then you have to tap and hold any one of the app until they start to toggle with a little red cross on them. Then you have to press that red cross to close that app which are using your iPhone’s resources (RAM, CPU etc.).
In iOS 7 you still have to double tap the home button to bring the list of currently running apps, but now you can see the screenshot of the last used the apps that are running in the background as well the icon of that app under the screenshot. You can Swipe left and right to scroll through the list of apps.
To close an app you have to swipe the screenshot upwards to close it. It is much easier and more feasible than the previous versions of iOS as you can swipe with two fingers and close two apps simultaneously.
It feels quite similar to Android where you have to swipe towards left to close any running app, but it feels good to have best of both worlds ;-)