In this case, everything with your car's alternator seems to check out OK — no belt issues or other visible signs of trouble — but the car's battery is dead, as is most of the car's electrical equipment. That might mean that the alternator is producing electricity, but it's either not going anywhere or it's not the right type.
Alternator electricity is piped through large cables and smaller wires. Any problems within the wires, cables or connections at either end can reduce or stop electricity from getting through. Occasionally, a symptom of this issue might be brighter lights, as the alternator produces more energy to overcome the resistance in a bad wire or a broken or loose connection. This symptom is usually accompanied by the smell of hot wires, too. Higher resistance within a wire creates heat, similar to the way an electric burner on a stove heats up by resisting the flow of electricity.
Another problem could be the alternator's diode rectifier. Alternators produce alternating current (AC) electricity in three phases, but a car's accessories require direct current (DC) to operate. The rectifier changes the current from AC to DC. Without that critical component operating properly, the electricity the alternator produces can't be used.