Charger issues - seem to be computer related (not charger)

My computer died while I was using it so I plugged in the charger but this did not change the charging status on the computer.

I thought it was the wall socket the charger was plugged into, but that was not it. I learned that when I had only the wall socket plugged in the "working" light glowed and when I plugged the computer end of the charger into the computer's charger port, the "working" light on the charger would shut off, indicating the circuit was broken.

I tried plugging in the computer end first then the wall end, but it made no difference, the light would shut off and my computer was not able to charge. This leads me to believe the issue is with the computer and not the charger. Help?

Also I am not tech-savvy so please spare me jargon. I just need to know if I should take it to a computer repair shop or try a new charger or new battery or what. THANK YOU!