Hi,
You can try to measure with a multimeter the 5V pin after you plugged in your external power supply (adapter or 9V battery). Please search a tutorial because if you do something wrong, your board can be ruined. Also you will find it very usefull if you are new with arduino and electronics in general.
Before pluging the power supply, test the program runing on Arduino with the USB adapter or cable to see if the problem is in the code. If your board doesn't do nothing, please try to upload a simple program with an LED (blink or On-Off) and don't test the power supplies only if the code is flawless.
Also, please upload a picture with the arduino, the power adapter jack and the 9V battery jack, because I suspect that can be a problem with the jacks in your case, especially that when you are on USB your board seems to work. I think your problem is with the power supplies jacks that are not workin