If the sight of a failing alternator isn't obvious enough for you (and it's totally understandable if it's not), the sound might clue you in. Drivers will sometimes complain about hearing a "growling" or "whining" noise before an alternator gives out. The alternator is driven by either an accessory belt or a serpentine belt in conjunction with the crankshaft pulley. The alternator pulley typically spins about two or three times faster than the crankshaft pulley to produce the power necessary at lower engine speeds, such as at idle. The alternator pulley spins on a shaft, which in turn is supported by either bearings or bushings. If the pulley is not in correct alignment with the belt, if it's canted on the shaft or if the bearings and bushing are worn out, the growling or whining noise will let you know there's a problem.
Since lots of car problems cause strange noises, this symptom is most useful if it is paired with other symptoms, especially since it can be hard to identify the source of a sound while you are driving.
If the alternator is so far gone that it has caused damage to your engine, you might hear rattles from inside the engine caused by failed engine bearings. Engine damage is one of the worst-case scenarios of alternator trouble and it is definitely a possibility, but it's so difficult and inconvenient to keep a car going with a bad alternator that you'll probably get it fixed long before it causes damage to your engine.