Well most phones will show a red charger light even when a battery is out. Just meaning there is no charge/needs charging. Obviously if the battery is out, then the phone needs power/charging.
Now if you have used your phone until it was very drained, battery at 0%, then it can get 'stuck' and not want to charge. To get past this try making sure the battery is in it, then press and hold the power button for 25-30 seconds. Release the button and plug in the phone to charge. You are not trying to turn it on, just to get it unstuck. Now leave it to charge, and turned off, for at least 3-4 hours, or the light turns green. Once that has happened, take it off the charger and try turning it on.
Now if it doesn't charge up the battery, you may need to test the charger cord, try it on a different device, to make sure it is working correctly. If that isn't the problem, then you may need a new battery.