1. Reboot methods. The first thing to do is try to reboot the iPhone, which you can do by holding the top button and the home button for about ten seconds. In some cases, the iPhone is actually still on, but a bug stops the screen from displaying anything, so you can reverse the problem by doing the reboot. If you still see nothing on the screen, try to plug the iPhone in and again try the reboot methods to see if you can get the screen to display.
2. Restoring the iPhone. If the iPhone shows up in iTunes, but it still doesn't seem to turn on, it's definitely a display problem. When rebooting fails, you can restore the iPhone from iTunes, and this should fix whatever bugs caused the original issue. Realize that you'll have to reload your music if you restore to another software version. To restore, plug the iPhone in, go into iTunes and click the iPhone's icon on the side of the select screen. Click on the restore button and agree to the prompt asking if you really want to restore the iPhone's software.
3. Recovery mode. You can put the iPhone into recovery mode if it has a charged battery by holding down the home button before connecting it to iTunes, then continuing to hold the button until iTunes recognizes that the iPhone's in recovery mode. Recovery mode is a bare-bones way of booting the iPhone that allows you to perform a restore when you're dealing with a serious issue such as the iPhone failing to turn on or display anything on its screen.
4. Last resort options. If you still can't get any response from the iPhone, you may have a bad battery or shorted out electronics. This is far from an ideal situation, but if you've got a warranty open with Apple, it's covered. If you don't, you can send the iPhone away for repair, but it will cost you a bit to replace the damaged components. Stop by an AT&T store and they may be able to give you a loaner phone in the meantime so that you don't lose your ability to make and receive phone calls until your iPhone is repaired.