The laptop power supply is not getting it's power to the inverter because the fuse that allows this is open
the power from the battery is able to reach the inverter and light up the cold cathode lamps that light up the screen because it's fuse is good however; the power supply to the inverter fuse is bad this fuse is very tiny and is located on the motherboard
to repair this is very hard and, expensive so; just use the laptop with the battery

Lots of laptop have power profiles that switch when either AC is plugged in or removed.
There should be a memorized screen brightness setting in your power management options. Usually it's located in the built-in Windows settings under Control Panel -> Power Options -> Edit Plan Settings.
You should see a brightness slider for the screen for "On Battery" and "Plugged In". Pick your brightness from there and it should stick.
Another way is to use the "FN" key on your keyboard. Hold down the FN key and press the associated brightness picture, which on your Acer, should be the RIGHT and LEFT arrow keys. Keep holding the FN and tap the right or left to increase or decrease the brightness. Usually the setting will stick this way, but somethings you have to use the control panel method for it to really stay.
You also might have another program for power like Acer ePowerManagement which should have the same settings as windows, but layed out differently.