Your alternator is the part of the car that charges the battery in your car. It does this by taking the energy that you create from actually driving and using this to recharge the battery which powers things like the engine and the lights as well as the ignition which starts the car.
This then means that without an alternator you would find your car ran out of battery power and was then thus unable to operate at all – you wouldn't be able to turn on the lights or listen to the radio, let alone actually start the car. As such then, this exactly what you would find if your alternator stopped working and you would no longer be able to do any of those things. However the main difficulty with alternator problems – and knowing whether or not to replace it – is that these symptoms can be caused by a faulty alternator or a faulty battery.