If you've found your iPad refusing to turn off, endlessly spinning the progress wheel, or worse, refusing to turn on, offering nothing but a dead black screen despite the fact that you know it's fully charged -- don't lose it just yet. Your iPad may be fine, and just have had iOS or an iOS application crash.
Before rushing to the Apple store with your iPad, try holding the home and sleep buttons together to restart/reboot your iPad. Be sure to hold the two for at least 10 seconds, perhaps longer.
In most cases, seemingly dead iPads will reboot normally (the Apple logo will appear).