Is your home button completely unresponsive? In another tip, I reccomended a method of cleaning out the dust and dirt that can accumulate behind the iPhone home button to fix unresponsiveness (won't return to home screen or bring up multitasking).
Unfortunately, sometimes this trick doesn't always work, or in some cases, the home button itselft is broken. In these cases, you can bring up the on-screen home button - what apple calls "Assistive Touch".
To turn on Assistive Touch, from the home screen:
Settings > General > Accessibility > AssistiveTouch
Once enabled, you'll see the assistive touch "button" on the edge of your screen, which you can move around as needed. When you tap on it, it brings up a menu that includes a virtual home button that you can then use as you would the physical home button on your phone.