Diagnosis
This is almost always a screen timeout or a setting that's been accidentally changed. The green power light means the parent unit is on, but the display is off.
How to Fix It
First, just press any button on the parent unit, like the volume or talk button. The screen should wake up. If that doesn't do it, the screen brightness might have been turned all the way down. Press the "Menu" button, navigate to the screen brightness setting (it's usually an icon of a sun), and increase the level.
If That Didn't Work
Try a full reset. Hold down the power button on the parent unit for a full 10 seconds to force a reboot. It should power down, then you can turn it back on normally.
Heads Up
If the screen only works when it's plugged into the charger, your rechargeable battery inside the parent unit might be worn out and need replacing. They do lose capacity over a couple years of nightly use.