Use an inexpensive digital voltmeter to measure the battery voltage. It must be 12.6 volts. If you find closer to 12.2 volts, it is good but discharged. If it's around 11 volts, it has a shorted cell and must be replaced.
Charge the battery if necessary, then once the engine is running, measure the battery voltage again. It must be between 13.75 and 14.75 volts. If it is low, suspect the generator. If the generator is defective, replace the battery too unless it is less than about two years old. These generators develop huge voltage spikes that can destroy the internal diodes and voltage regulator. As the battery ages, it loses its ability to dampen and absorb those spikes, leading to repeat generator failures.