Most of the actual theory behind Buddhist reincarnation doesn't have to do with "souls" - one of the basic Buddhist concepts is one of anatta, or no-soul.
Likewise, most people's idea of reincarnation is totally bunk when compared to the actual theory. Think constant changes in the idea of a "self", of which there is no actual self other than one's conception of it and this is mentally and spiritually reborn at any given moment - dies and then is renewed. Most people just think of physical death, because their concept of reality is only what they can sense. Likewise these labels and preconceptions of the self are passed on through thoughts, actions, and social norms. It's the point where this realisation of the not-self occurs that enlightenment and understanding of the nature of happiness and reality is reached.
Nirvana, or Nibbana in the Buddha's language, literally means "extinguishing of the flame" - and has to do with the metaphysics behind how the ancient Indians viewed what happened to a fire once it goes out. What could be called a flame is now gone, but the energy is still there, dispersed, all around us. Just the fuel, the agitation, the desires we have that keep the fire burning - those have been put out.
That being said, it's all phenomenology. We pay way too much attention and lip-service to external forms. Those aren't so important in the scheme of things.