What is my phone doing?

I just pulled my iPhone 4S out of my pocket and it was scathing hot. My first assumption was that the GPS is stuck on. I checked it out, and the GPS is not being used.

I decided something must be running in a background process and that I should shut my phone off to make sure it was killed.

When I shut the phone off (press power and home at the same time, then slide the red slider over to power off), the screen goes black with a little spinner, and then the phone boots back up automatically. I literally can not shut my phone off. I've tried 3 times now and it automatically boots itself again.

Also, my battery is half drained. At 2pm. Half.

What is it trying to do?