Simple test. Hook a DVM to each of the battery terminals. Record the voltage with the car off. Record it then again w/ the engine running.
If the voltage is higher, then the alternator/regulator is working likely.
Next pop the vent caps to the battery. Make sure the water is full in each cell, up to the bottom of the plastic case. You dont want it to be flush with the top of the holes, that's overfilled. So with the engine off, put neg. test lead to the neg at the battery. Then stick the pos. test lead in the water of the first cell closest to the neg terminal. You should read 2.2V. Then go to the next cell, and you should read 4.4V, etc etc.....
Each cell is ~2.2V, and the are wired in series, so each cell adds 2.2V to a total of 13.2V or so on a fully charged battery. Now if one cell doesn't add any voltage to the total count, you have a bad cell.
Another thing to mention, with the car running , measure the voltage at the battery. Should be initially 14.4 or so, then drop down slowly. Once it reaches around 13.6 or less, turn the car off. Then measure the voltage again. It should be slightly less, but maintain above 13V for a while. If it drops down drastically, it's not taking a charge. maybe the acid is gone, or something. Bad battery, time to swap.
note: this should be done with all loads off. That includes the dome light and the hood light. No amp draws should be present, that includes your stereo and such.
Scott