The picture of Platonic souls we picked up from the Phaedo may be misleading. In that dialogue, Socrates is particularly concerned with comforting his followers about his death. He touches on knowledge and eternity, but says very little about the qualities and operation of the soul. The Republic gives us a deeper, if still somewhat obscure picture. Here the soul is neither unitary nor ideal, but rather distributed.
In science, we have come to think in terms of discrete objects and mechanical processes, but Plato wants to speak of principles of operation and aspects of the whole. Consider the spirit of cooperation necessary for churches and sports teams. We are well aware of this as a property of the ensemble. We can speak of qualities and functions without localizing them. This is not to say that a team is not composed of members, only that this property is best analyzed at the team level. For a scientific metaphor, you might think of balmy air. In some way, the “balm” results from the aggregate speed of nitrogen molecules (air temperature) and the density of water molecules (humidity). Nonetheless, the property “balmy” is more descriptive and useful 99 times out of 100. For Plato, the soul is the operation of a living body rather than an operation of part of the body.
Plato wishes to address the possibility of conflict within the soul. How can we be divided against ourselves? How can we will two things at once? This should be a familiar experience for most of us. The word “will-power” has come to mean the triumph of one of our desires (usually the long term one) over another (usually the short term one). We want another drink, but know better. We want to stay in bed, but want to keep our job more. Plato wants to account for competing preferences, as well as the complex interaction of body and soul. He wants to speak of the soul in a more complicated way than in Phaedo, both more integrated with matter and less monolithic.
To handle conflict, Plato invokes three aspects of human souls: appetite, spirit, and reason. The appetite is moved by bodily needs and desires: hunger, thirst, material and sexual wants. Plato sees these as good, contributing to the well-being of the body, but only when governed by the other aspects. The spirit is moved by honor, an abstract concept Plato framed around the esteem of our neighbors. We have courage when we overcome our appetite (say, fear of death) for the sake of the community. We have anger when other people frustrate our material desires (say, stealing our food). Spirit covers a range of non-bodily virtues, but can also cover vices when not governed by the third aspect. The reason is moved by wisdom, truth, and knowledge. Here the good and wisdom are pursued by the individual in the context of the community. Reason allows us to think critically about esteem and desire so that we may work for the greatest good.
It should be clear from the description that this model for the soul does not map easily onto modern concepts of mind, brain, consciousness, or will. It takes into account ideals, emotions, and even reflexes. This soul is not so easily separated from the body (as in Phaedo). Plato suggests that the highest aspect, the reason, is most amenable to philosophy and immortality, but all three qualify as soul, a trichotomy that Aristotle will develop more fully.
The ambiguity allows us to speak of souls as unique to humans (in their rational aspect) and common to all life (in their appetitive aspect). In both cases, the soul is the operation of the living thing, but only in humans does the proper operation include reason. [Some animals have spirit, he says.] Thus Platonic souls can be seen to bridge the gap between ideal and physical reality in an interesting, if not fully developed, way. Unlike the eternal unity of the soul in Phaedo, this psychic diversity in physicality was not handed down to the current age. It was picked up by various Christian theologians over the centuries, but never became a major point of doctrine.