I have a uno r3 about 2 weeks and every thing works fine on usb connection. After disconnect of usb and connect 9 volt battery the power light (green) comes on . The yellow light will blink a few times and than nothing .I have pressed the reset button a number of times but after the same sequence nothing changes. What is supposed to happen. It will not run sketch that I had just run with usb power. As an inexperienced person I don't know what to expect. 
Any help on this will be greatly apprehended. thanks "Dukie"
What pins did you connect the battery to?
For 9v, you should use VIN.
As an inexperienced person I don't know what to expect
You should expect the last sketch you down loaded to run when you apply external power.
Make sure that the battery voltage is not dropping below 7V when it is powering the arduino. Those 9V batteries are rubbish you know.
I am using the barrel connector on the board and 9 volt batt. that came with the kit . What is the barrel connection for? I was under the impression it was to power the board when disconnected from the PC. I will connect to Vin pin. Thanks for your reply. This is a wonderful resource and community to be in. I'm sure this not the last question I will have. "dukie" :~ in Orlando Fla. Thanks all.
What is the barrel connection for?
For powering the board.
I will connect to Vin pin
That will also power the board but there is no reverse polarity protection. But the up side is that your battery can drop another 0.7V before the arduino starts misbehaving due to low voltage.