Step 1-  double-click the home button. In iOS 7 and up, that causes the apps to fall back a bit so that you can see the icons and screenshots of all running apps. 

In iOS 6 or earlier, this reveals a row of apps below the dock.

Step 2 - Slide the apps from side to side to find the one you want to quit.

Step 3 - When you find it, how you quit the app depends on what version of the iOS you're running.

In iOS 7 and up, simply swipe the app off the top edge of the screen. The app disappears and it has been quit.

In iOS 6 or earlier, tap and hold the app until a red badge with a line through it appears. 

The apps will wiggle like they do when you're rearranging them. When the red badge appears, tap it to kill the app and any background processes it might be running. When you've closed all the apps you want, click the home button again to return to using your iPhone.

In iOS 7 and up, you can quit multiple apps at the same time. Just swipe up to three apps up the screen at the same time. All the apps you swiped will disappear.