Why isn't the iphone charging?

When you connect an iPhone to a power source, there should be a chirp sound indicating the iPhone is connected.

http://www.apple.com/iphone/how-to/#basics.charging-the-battery

Your wall charger is defective, go exchange it.

You're welcome.

Does the computer need to be on to charge?? (Stupid question?)

Yes - when your iPhone is connected to your computer, your computer must be on in order for your iPhone to charge. When your computer is on with your iPhone connected, if your computer is set to sleep after a period of inactivity, charging for your iPhone will stop, and your iPhone's battery will be depleted faster while remaining connected to a computer that is asleep. I have my MacBook Pro set to never sleep with only my screen set to sleep after a period of inactivity. This way I can leave my iPhone connected to my Mac without power being disabled to the USB ports.

Since your iPhone charges when connected to your computer but doesn't charge when connected to the wall charger, as already provided it sounds like your wall charger is defective.

Apple handles all iPhone repairs or exchanges under warranty, and everything included in the box is under the one year warranty for the iPhone. You can call AppleCare or make an appointment at an Apple store to have your wall charger exchanged under warranty.

Ditto here...the little white wall charger only intermittently makes the phone "chirp".
Is this a common problem? Should I go to an Apple Store or an AT&T store? (it was purchased at an AT&T store, but I really don't want to drive to THAT one...)

Apple handles all iPhone repairs or exchanges under warranty, and everything included in the box is under the one year warranty for the iPhone. You can call AppleCare, or make an appointment at an Apple store if there is one nearby to have your wall charger exchanged under warranty.