As a former Automotive Battery and Electrical Technician perhaps I can shed some light here.
Generators have residual magnetism, alternators do not. Score one for generators. However, generators do not put out enough juice at idle to say so, whereas alternators do. Score one for alternators. Greater output at almost all RPM's is why OEM's went with alternators to begin with (that, plus alternators are lighter). Chrysler (the inventor) started out with alternators in their top-of-the-line and bottom-of-the-line cars in '61, the Imperial and the Valiant. The test went so well they went with alternators across the board in '62. Every other OEM followed very soon thereafter.
Since alternators have no residual magnetism, it is possible to reverse your battery's polarity if you let it go dead. Incidentally, it's impossible for a charged battery to freeze, as the Specific Gravity is around 1.280, but a discharged battery (Specific Gravity near 1.000) can freeze overnight.
Jim Anderson is right, an alternator is self starting, provided you have at least a modicum of juice available to excite it initially. A good-but-dead or a bad battery doesn't cut it. He's also right on the alternator light as being essential; it's all part of a series circuit. The bulb must be good and working properly.
dpetryk is also right; your alternator should start charging the second it starts spinning.
I must respectfully disagree with Mike Murrell, however. Although my car is a Benz, my truck is a GMC 6.2 L diesel and I have had an Auto Zone Lifetime Warranty alternator on it for several years with excellent results. I forked over some extra bucks to get a 94 Amp unit instead of the OE unit which was 78 Amps. My training and experience both tell me that you can never have too big an alternator, especially in this part of the country! The way Auto Zone has treated me is A+. Considering how long I keep my vehicles, that Lifetime Warranty is pretty sweet!
I have found that a perfectly good alternator is often blamed for problems caused by a bad battery. To accurately test a battery it must be fully charged. Poke each cell with a Hydrometer and look for 1.260 to 1.280. This will tell you if all six cells are taking a charge evenly. If it's still in the game at this point, then you need to have a shop put a load on it and test its output voltage while under load. Then, and only then can you tell what kind of battery you're actually dealing with. Remember, to work properly, a battery must do three things: accept a charge, hold a charge when not being excercised daily, and give up its charge when needed. When a battery can only do one or two of these tasks, it's time for a new Die Hard.
A generator produces direct current, while an alternator produces alternating current which is changed to direct current by means of an internal diode rectifier. When your alternator idiot light comes on it usually means that one or more diodes are shot. Often this is done through careless or improper battery jumping, while either giving or receiving a jump start. A split second is all it takes to start frying diodes.
Hope I haven't confused you too much, Josh! Best wishes.