Why won't my iPod turn on?

You just likely need to replace the battery. Many devices will function with a bad battery while being connected to a charger. Once disconnected from the charger the device powers off because the battery does not have enough power to keep the device on and is no longer receiving power from the charger. Not sure what model of iPod you have but there are many guides on this site. There is likely one for your model on this site that could guide you through replacing the battery yourself. If not YouTube may have some tutorials. So I definitely recommend replacing the battery.