Hi, John! Welcome to iPF!
You can close them this way:
In iOS 6, tap and hold an app icon until they all start to wiggle. Delete them the way you would remove apps from your iPad.
In iOS 7, slide the pictures above the app icon towards the top of the screen.
This is called force closing apps.
The apps in the multitask bar (that's what you see when you double tap the Home button) simply indicate which apps you opened or used recently. Most of them won't even load or keep on working in the background. They are in a suspended state. Usually you don't need to close them, just from time to time, if you're low on memory (RAM), or an app doesn't respond. Leaving them in the multitask bar is also very useful, because it allows to change between apps without having to press the Home button every time. Just perform a four finger swipe to the left or right, and you'll find the app you used before.
Here's some more information about multitasking:
http://support.apple.com/kb/HT4211
Hope that helps.