When you double-tap the home button of your iPad or iPhone, the dock will show a list of all the recently used apps and some of them could still be running in the background.
You tap and hold your finger on the app icon until it begins to jiggle and you can then press the red “minus” sign  to close one or more of these background apps. If a particular app is not running, this action will simply remove its listing from the multitasking bar.

Close All Background Apps at Once

If you would like to close all the background apps at once, here’s a free app for iPhone and iPad that can help. You don’t even have to jailbreak your device.
Launch the app and then shake your iPhone / iPad to terminate every single app that could be running in the background. Alternatively, you can just pull and release the processes list  to close all the background running apps at once.

Should You Quit Background Apps Manually?

If you are on Windows or Mac, you always quit applications that you are not using so that more memory (RAM) becomes available to front-running apps. However, you many not see any improvement in performance by closing background iOS apps.
Here’s an excerpt from a Q&A that Engadget had with Steve Jobs:
Engadget: How do you close applications when multitasking? 
Jobs: In multitasking, if you see a task manager, they blew it. Users shouldn’t ever have to think about it.
John Gruber explains why closing background apps isn’t required:
The iOS multitasking bar is not like the command tab switcher on Mac or Windows. It is not a list of currently “running” applications. It is simply a list of your most recently used applications, whether they’re running in the background, suspended in memory, or completely inactive.

Exception – Kill the Misbehaving Apps

In the past, I have seem apps, especially games, that became unresponsive (freeze) andforce-quit was the only way to get them working again. The Process Killer app mentioned above might be helpful in such situations though else you need not worry about shutting down suspended background apps.