Since the alternator supplies your vehicle's electrical needs, when it begins to lose its potential, so do the car's accessories that draw on that electricity. Your car 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 apparent reason.
Your 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 your 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, your car's heated seat will turn off or the radio will quit before the headlights dim and fade away. That's because you need to be able to see in order to safely pull over and stop if your car suddenly dies — your radio and heated seats working simply become secondary at that point.