Common signs of a failing alternator include an illuminated warning light on the dashboard, dimming lights during the vehicle's operation, and a constantly dead battery. Many auto parts stores can test an alternator for free to determine if it needs replacing.
An alternator's function is to use the rotating movement of the engine to generate electricity to charge the car's battery as well as power the running electrical systems. If the alternator's output goes below a certain value, usually 13 volts, a warning light on the dash illuminates. This light is either in the shape of a battery, or it is labeled as "ALT" or "GEN."
Since the alternator also powers the electrical system, when it's failing the headlights and dash lights start to dim because the alternator can no longer handle the extra load of these accessories. This is especially true if the brightness fluctuates as the accelerator pedal is pressed.
When the alternator is bad, the battery ends up working harder to power the vehicle. Since batteries aren't meant for long-term use, they eventually lose charge and the driver is left stranded. That is why a test for the alternator involves jump-starting a dead car; if the jumper cables are removed and the vehicle dies shortly after, it means the alternator isn't producing enough power to maintain the vehicle and charge the battery.