Your vehicle has a number of different warning lights dedicated to alerting you of potential trouble, and one of the more uncommon is the alternator warning light. Unlike the "check engine" light, which seems to be triggered by just about any issue, the alternator light will likely not be one that a driver sees very often, and thus you should pay close attention if you notice it has come on.
First, it helps to understand what an alternator is. The alternator is tied into the electrical system of the vehicle, and is probably the most critical component to keeping the car powered besides the battery itself. Car batteries are said to have a very long lifespan, but this actually refers to the length of time before they stop holding a charge. Your vehicle's battery needs to be charged like any other electrical device you own, and it's the alternator's job to provide that juice.
The alternator is attached to the engine's crankshaft, which rotates as the car moves in order to generate mechanical energy. The alternator then turns this into electricity that can then be passed on to the battery. This is why you could essentially drive for miles and miles with your headlights on and the stereo blaring with no battery issues, but leaving the lights on overnight can sap your charge - the crankshaft isn't turning, so the alternator isn't doing anything to provide more power to the battery.
In this way, think of a malfunctioning alternator as a similar situation to losing the charge cord for your cell phone - you'll be able to keep using it for a little while, but it's living on borrowed time. Once the alternator warning light turns on, your best bet is to try to find a garage or auto parts store before your battery completely drains. Fortunately, the battery should last for a few hours after the warning light comes on, depending on your electricity usage. Running the fan, stereo, interior lights and headlights will obviously drain it much quicker than going without those items for the time being.
Of course, as with any warning light, there are occasionally false positives. Sometimes the light may come on and the issue is actually with the light or the gauge itself rather than the alternator. An easy way to test this is to put the car into park and turn your headlights on. Then rev the engine by pressing down on the pedal. When the alternator is having an issue, the headlights will either brighten or get dimmer as the engine revs.
On the other hand, if the headlights shine with the same intensity, it's probably an issue with your warning light - something you might want to have a mechanic look into, but not a problem that requires your immediate attention.
Legitimate problems with the alternator vary in their severity. One of the more common issues is the alternator drive belt simply slipping off the pulley that it's intended to be attached to. If the belt isn't broken, this isn't a very complicated or expensive fix. Another issue could be the alternator itself wearing out, which will probably be due to one of the diodes on the part being overcharged with electric current.
Finally, as with almost all electrical problems, the issue could be related to faulty wiring, which likely will not be an expensive repair.