Can't I just press the "power button"?
No, because this button is actually a "Sleep/Wake" - pressing this button puts your iPad/iPhone/iPod Touch to sleep. It does not power off the device, which is why you can have background apps, such as the alarm, still able to work. Unfortunately, Sleep Mode will typically not help a frozen iPad, or clear up any strange problems.
iPad Home and Sleep/Wake buttons indicated
Option 1: Restart your iPad
Starting with the easiest of the three options, this lets you truly power off (not just put it to sleep) and then restart your device.
1. Press and hold the Power/Sleep button until you see the red arrow and the "Slide to Power Off" pop up.
2. Follow the instructions and slide your finger to power off the iPad (alternatively, you can tap the Cancel button to get out).
3. Then, to restart your device, press and hold the Power/Sleep button until you see the Apple logo.
After the logo goes way, the restart is complete and you can get now start up your iPad normally, getting to the start screen by pressing the Home button or the Sleep/Wake button.
Option 2: Reset your iPad
If restarting your iPad doesn't solve the problem, resetting your iPad probably will. Don't be scared off by Apple's "reset" terminology - this is simply the equivalent of an "Alt+Ctrl+Delete" to clear a problem. It is not a factory reset or anything close; none of your settings, data or apps will be disturbed by this "reset".
1. To reset your iPad, simultaneously press and hold the Home button and the Power/Sleep button for 5 seconds or so, until you see the Apple logo. Continue holding if you first see the "Slide to Power Off" pop up until the Apple logo appears.
2. The logo will stay for 20 seconds or so and then revert to your start screen. At this point the restart is complete.
3. Start your iPad normally, pressing the home button or the sleep button to get to the start screen.
Still Dead? Option 3
If the above two options don't work as your iPad is completely unresponsive - nothing happens when you press the Home button or the Sleep/Wake button - don't panic, just recharge your iPad. Most likely the battery is completely dead. Be forewarned (don't worry) when the battery has been drained it still may take several minutes after starting the recharge before you can tell your iPad is still alive.