iPad turned itself off and won't turn back on I was just reading a webpage on my iPad and the screen went black and then the Apple logo came up and then the spinning loading wheel and it just went black...

Nothing after that.

The screen is blank/black/off - can't tell which. Resetting does nothing. Plugging it into the charge does nothing. Plugging it into iTunes does nothing.
That sounds like behavior consistent with a dead battery. How long did you leave it plugged in?

I'd plug it in and leave it charging for 15+ minutes, then hold the home button and power button simultaneously for 10 seconds to do a hard reset. If that doesn't work, it's off to the Apple Store to get a new one.