Does the battery icon show that it's charging (lightning-bolt icon)?
If not, then be sure your charger is plugged into a working power outlet (plug something else into that same outlet). If you have another cable, try another cable. If you have the "charger dock thingy" plugged into a power outlet, try plugging it into the USB port on a computer.
If the icon shows that it is charging but the battery drains anyway, try restarting the iPod - full power-off (Slide to power off). Try force-quitting the apps that are currently running (double-click Home button, slide app "pages" off the top of the screen) and restart again.
If none of that helps, consider backing-up the iPod and Restore iOS (erase and reinstall the operating system), and set it up as if it was a new iPod. If it works well, then restore your backup. If the problem comes back (which is rare), then there's probably something wrong with an app and/or your user data.
But in the end, if your iPod is less than a year old, it is still covered by warranty. Contact Apple now, don't wait until after the warranty expires. They'll have you do everything I mentioned above, but they can also test for a defective battery.
