Did you check the alternator belt tension? I would do that first. After that you will need to get a digital volt meter (they are easiest to read). A fully charged battery should read about 12.5V or even a little higher.
If it does read 12.5 or so, start the car, then check the voltage again across the battery. Should be about 1.5 volts higher to charge the battery, if not and the alternator belt is tight, then either the regulator or diodes(inside the alternator housing) are toast, or the alternator itself has a problem.
All of the above, besides the belt, indicate the alternator will need to be removed and bench tested, and most likely repaired.
If it does not read 12V or higher the battery needs to be load tested, to see if it is still good, unless of course it is new
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I would recommend you also check the terminals for tightness. especially the ground lead, since they tend to loosen, although if the car starts fine, this is less likely then the other things above.
HTH,
LarryB