How old is your battery? If it's more than 3-5 years old, then it might be bad.
The battery light signifies low voltage, which usually means the alternator isn't doing its job if the car is running already. If your alternator is functioning properly (making 14+ volts), then there is enough current to keep the car running even if the battery itself is dead. The battery is mostly there for starting the car.
Without the alternator providing charge, the battery would run down to the point where it could no longer give spark to the engine or turn the starter motor. Running down like this can kill an older battery as well (had this happen to me once), to the point where it would no longer hold a charge even when jumped.
It's possible one or both of the alternator or battery are toast. I wouldn't suspect the starter, especially if you're getting the low voltage light.
Have a mechanic at an auto shop check it out. Even a tech at an auto parts store should be able to check the battery and alternator to make sure they are functioning properly.
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