Well the concept of an automatic battery backup is pretty straight forward, the details and added features (like auto recharging or auto power down on low battery) one might want can take some extra thought.
The easiest way to implement automatic battery back-up while normal power is being supplied to the Arduino external power jack is to wire a battery with greater then around 8vdc to the Vin pin and ground pin, with a series isolation diode (1N4001 is fine) between the battery positive lead and the Vin pin, with the cathode wired to the Vin pin. With a two resistor voltage divider you can also monitor the battery terminal voltage via a analog input pin.
PS: This is probably not the best sub-forum to post this topic.
Lefty