A Dead Battery
While not technically an alternator problem, a car that won't start is a definitive notice from your car that there's a problem in the starting system. When the key is turned, you'll hear a tell-tale clicking of relays ticking over but nothing else happens. After a time, even that noise stops. The immediate problem is usually a dead battery; but the question a driver should ask is, "Why is it dead?"
When an alternator begins to fail, or fails outright, the car's battery begins to take up the slack rather than acting as a capacitor for the system by receiving a constant infusion of electrical power from the alternator. However, even the best car battery will run down eventually leaving a driver stranded in the driveway or worse, on the side of the road. Car batteries are not designed for long-term power use. They're designed to provide the vehicle with enough electrical oomph to start by juicing the starter motor with a surge of power and getting the whole works spinning.
Diagnosing a dead battery versus a dead alternator is relatively easy. Simply jump-start the car and then remove the cables as quickly as possible. Then wait. If the alternator is failing to charge the system, the car will soon die again and the problem has been pinpointed. If the car runs and continues to run, then it's likely the problem is with the battery. Caution should be used, however, as a dying battery can only receive a charge for so long and may go completely dead at a later point despite the best efforts of the alternator. Testing a battery can be accomplished with a voltmeter, and most auto stores have the facilities to check a suspect battery's state of charge and even its charge capacity.