Since the alternator supplies the vehicle's electrical needs, when it begins to lose its potential so do the accessories that draw on that electricity. Drivers may begin to experience erratic symptoms ranging from dimming or extremely bright headlights and dash lights, to speedometers and tachometers that simply stop working for no reason. Other accessories, like heated seats or power windows may experience a slowdown as well.
A driver's exact experience is usually dependent on a number of factors. The first is how well the alternator is still producing power and also where it is in its death cycle. The second is how the car is programmed. In most new vehicles auto manufacturers have a sort of preprogrammed priority list for where electricity will be sent just in case an alternator problem arises. This is usually based on safety considerations. For example, the heated seat will switch off, or the radio will quit before the headlights dim and fade away. The reasoning behind this is that a driver would need to be able to see in order to safely pull over and stop if the car suddenly quit working -- but they definitely wouldn't care about a heated seat or what's on the radio at that point.