All in all, this is not great news. In the best case scenario, it's a huge inconvenience, but in the worst case scenario you're going to need to replace the alternator and possibly the fan belt, which can be hugely expensive.
If the battery or alternator light is lit up, it basically signals that there is more electricity being utilised by your car than is being replenished. You may get some warning signs before the light comes on (or at the same time as the light comes on) that you may want to keep an eye or ear out for. The audio on the radio fading in and out is one sign that the electricity in your car is running low, as well as the sounding of the horn and the brightness of the headlights decreasing.
The number one thing to remember if the alternator light comes on is to keep driving until you reach the next service station. If you stop the car, there's next to no chance of starting it up again. It will also help to turn off the radio and any air conditioning facilities, and unplug any devices, such as a phone charger; all of these things are going to leach your car of electricity.
The tricky business with alternators is that the rules are different if the temperature gauge lights up at the same time the alternator light does. If this is the case, the reverse to the above applies: you should stop the car as soon as you can find a safe spot to do so. This is because the fan belt that controls the alternator is the same fan belt that controls the water pump - meaning if there is a problem with the fan belt, the water pump is going to seize and the engine will become so hot that its parts will begin to melt. Continuing to drive a car that is overheating can, and most likely will, do extreme damage and will cost a bomb to repair, so it's important to pay attention to what else is happening in your car when the alternator light comes on.
