Fundamentally, your problem is that he alternator is not charging the battery. A good alternator will put out 13.6~14.4V - your alternator is not doing anything, which is why the voltage is lower with the engine running.
Tim's and Mike's advice is bang -on.
First step is to check whether the charge light comes on when the ignition is on but the engine is not running. If it doesn't, then the problem is most likely either worn out brushes or the old 240 biodegradable wiring.
Pulling the brushes is literally two screws on the back of the alternator - if the brushes only hang out a couple of millimetres, then they are worn out. New brushes hould be about $10 from a competent parts place or competent auto electrician and take about five minutes to install if you have a soldering iron.
If the brushes are good, then check the globe in the instrument cluster. If it is good, then start chasing wiring.
If the charge light does come on, then check the voltage at the big terminal on the back of the alternator with the motor running. If it is different (higher) to what you have at the battery, then check all the connections, including the earths.
The above will get 98% of faults, If you are in the 2%, then report back what you've found and we can dig deeper.