There are many different ways of designing an alternator so it is hard to know where to start.
It is possible that the excitation circuit is faulty. This can involve sliprings and carbon brushes. In this case check for corroded sliprings or worn out brushes. I have successfully refaced coroded sliprings in situ using a flat bladed screwdriver while the engine was running.
Some generators have an electronic automatic voltage regulator (AVR). This is usually a block of epoxy with wires coming out of it. Often connected to sliprings. They are not normally repairable.
Those without an AVR or sliprings usually have a large capacitor. This is generally a cylinder often white in colour with spade terminals. It will have a number followed by the letters "uF". If the capacitor is short circuited or low in value the generator may not excite. If it is faulty be sure to replace it with the exact value. More then a few percent out can result in excessive output voltage and damage any connected load.
It could also be a burned out winding.