Polkinghorne briefly states his conception of the soul in the introduction to The God of Hope and the End of the World:
The soul ... is conceived of as the information-bearing pattern carried by the matter of the body, a revival in modern dress of the Aristotelian-Thomistic idea of the soul as the form of the body. It is claimed to be a coherent hope that God will hold that pattern in the divine memory following its dissolution at death, and then finally restore a person's full humanity through the re-embodiment of the soul in the final great act of resurrection.
Before we can understand this fully, we'll need to understand his philosophical outlook on the relationship between matter and spirit (which he terms "dual aspect monism") and what he means when he refers to an "information-bearing pattern."
One more note: Polkinghorne does not address whether or not this definition is unique to humanity, but it's worthwhile to note that Thomas Aquinas, to whom he acknowledges his indebtedness, believed that there were three kinds of souls: plant, animal, and human. Polkinghorne only seems to have humanity in view in his discussions of the soul, but he probably wouldn't rule out other beings having souls in a Thomistic sense.
Dual aspect monism
Philosophies of mind are broadly broken down into monism and dualism. Some are hard to classify as one or the other, including Polkinghorne's preferred theory, which is called "dual aspect monism" or "double-aspect theory". In an interview, he offers this description:
"Dual aspect monism" is an attempt to wrestle with the persistent unsolved problem of how mind and matter relate to each other. It differs from classical dualism, which maintains that there are two sorts of substance: mind and matter. Its problem was how they relate to each other. I'm sure that we're not simply matter, and I'm sure that reality is more than just ideas. None of the classical solutions seem to correspond to our experience. Dual aspect monism tries to take seriously both our mental experience and our material experience. It says that they're related to each other in a very deep and complementary way, that there is only one stuff in the world. Dual aspect monism seeks to avoid devaluing or subordinating one side or the other. Sometimes it might seem a little like a subtle form of materialism, but I don't think it is, because it doesn't treat the mental as being just an epiphenomenon of the material.
He gives the subject a much more in-depth treatment in his book Faith, Science and Understanding, pages 95-98. First he describes in what way classical monistic and dualistic conceptions have proved inadequate:
The classical metaphysical strategies of materialism, idealism and Cartesian dualism all exhibit a bankruptcy in the face of the many-layered, and yet interconnected, character of our encounter with reality. This recognition encourages the search for some form of dual aspect monism, an account that would acknowledge the fundamental distinction between experience of the material and experience of the mental but which would neither impose on reality a sharp division into two unconnected kinds of substance nor deny the psychosomatic unity of human beings. Stating this metaphysical aspiration is one thing; its attainment, even in a sketchy and conjectural form, is quite another. ...
Just as relativity theory has integrated matter and energy into a single account, so one might hope for an eventual discovery (at least as revolutionary as relativity, and most probably much more so) that would integrate the triad: matter-energy-information. That achievement would be a significant step in the search for a dual aspect monism. ...
It would, however, be no more than a first move in a direction whose ultimate goal was still over the horizon. A much more important step would have been made if it were ever possible to attain an understanding in which consciousness was taken into an integrated account. Despite many over-confident reductionist claims to the contrary, consciousness is presently both an undeniable experience and also an irreducible mystery.
With that established, he moves on to describe his own conception of dual aspect monism and what it entails:
If a dual aspect monism is on the right track, then there will be entities, such as stones, whose nature is located wholly at the material pole, and other entities, such as ourselves who are 'amphibians', participating in both kinds of polar experience. It would seem entirely reasonable to suppose that there are also entities whose nature is located wholly at the mental pole. (This could provide a metaphysical lodging place for created non-embodied spiritual beings, such as angels, if such there are.) There is another candidate for this end of the metaphysical spectrum, less controversial than angels, namely, the truths of mathematics. There is a widespread conviction among mathematicians (which I share) that the pursuit of their subject involves discovery and not mere construction. Mathematical entities, such as the prime numbers and the Mandelbrot set, are 'out there' in some platonic world of mathematical ideas. Not only can a dual aspect monism accommodate such a belief but also this conviction of the mathematicians would point to an interesting aspect of such a metaphysics.
The material world is a world of process, characterised by temporality and becoming. ... A noetic world containing the truths of mathematics would have a different character. It would be everlasting, in the sense that such truths just are and do not evolve. ... It follows then that, if these two worlds, material and noetic, are but complementary aspects of a larger created reality, then the duality involved in that wider picture is more than just that of material/mental; it must also embrace becoming/being and everlasting/temporal. Once again, humanity is the 'great amphibian', participating in both poles of this complex reality.
On pages 18-19 of The God of Hope and the End of the World, Polkinghorne attempts to illustrate the need for science to account for an idea of "information as a complement to energy in the description of the process of the world." To that end, he describes an experiment done by Stuart Kauffman originally described in chapter 4 of Kauffman's At Home in the Universe. Here is Polkinghorne's summary:
A physical realisation of his computer model would consist of a large array of light bulbs, each of which has its behaviour of being on or off correlated with the behaviours of two other bulbs somewhere in the array. If they are both on, it is more likely to be on also at the next step of the system's development. If the system is started off in some random configuration, with some bulbs on and some bulbs off, and then allowed to develop according to these rules, instead of just flickering away haphazardly for ever, the system soon settles down to cycling through a very limited number of particular on/off patterns. This unexpected behaviour represents the generation of an astonishing degree of overall orderliness. If there are 10,000 light bulbs in the array, there are about 103000 possible configurations that might in principle occur. In practice, however, the system cycles through only about 100!
This remarkable generation of order out of chaos strongly suggests that if the behaviour of complex systems is to be described and understood adequately, this task will call not only for the conventional 'bits and pieces' account in terms of the interactions of constituents but also for a complementary holistic account in terms of the overall pattern of the whole. In scientific terms, one can say that the conventional picture of energetic exchanges between particles will need to be supplemented by a description of the whole, framed in terms of the effects of information generation (that is to say, the specification of complex pattern).
We see here the prospect of the revival of an antique notion, reclothed in modern dress. Aristotle had spoken of both matter (hyle) and form (eidos). We are just beginning to learn how to speak, in a parallel way, about energy and information. Just as Thomas Aquinas used the revived Aristotelian science of his day as an aid to his theological thinking, so we, in our time, may find theological value in making use of the analogical resource that these scientific developments offer us. In pursuit of our present eschatological task, we shall see later that this method may be of particular relevance to how we may understand the nature of the human soul.
Definition of soul
We should now be ready to understand Polkinghorne's definition of the soul, which he gives on pages 105-106 of The God of Hope and the End of the World and pages 61-62 of Science and Theology. Quoting first briefly from the former work:
The soul must be the 'real me' that links the boy of childhood to the ageing academic of later life. If that carrier of continuity is not a separate spiritual component, what else could it be? It is certainly not merely material. ... What does appear to be the carrier of continuity is the immensely complex 'information-bearing pattern' in which that matter is organized. This pattern is not static; it is modified as we acquire new experiences, insights and memories, in accordance with the dynamic of our living history. It is this information-bearing pattern that is the soul.
Now quoting from the latter for a more complete definition:
A primary human concept is that of the continuing self, linking golden-haired youth to snowy old age. (At least, that is so in traditional Western thinking; Eastern thought has more frequently considered the individual self to be an illusion from which release is to be sought. In this book we shall take the Western view.) One of the attractions of dualist thought is that it assigns to each human person a spiritual component, a soul, which acts as the carrier of the self, defining a unique human identity in this life and beyond it. Yet the tenor of the foregoing argument has been to reject dualism and to treat human beings as psychosomatic unities, 'animated bodies rather than incarnated souls', in a famous phrase. This was the way in which the ancient Hebrews seem to have conceived of humanity and a psychosomatic account of human nature is the dominant, but not exclusive, way of thinking to be found in the Bible. One of the few matters relating to humanity on which there is a substantial degree of contemporary agreement is that men and women are to be treated as unities and not as spiritual beings housed in fleshly bodies.
This does not imply eschewing all talk of the soul - that would be a grave difficulty for much of theology - but a redefinition of what the soul might mean. In essence, it must be 'the real me'. It is clear that this is not simply the material that happens to compose my body at some particular time. The atoms in each of us are continually being changed by eating and drinking, wear and tear. They cannot be the source of our experience of a continuing self but, rather, we may suppose that the self is composed of the immensely complex 'pattern' in which that matter is organized. It is beyond present human power to explicate exactly what are the characteristics of this pattern, what changes (as new memories are acquired, for instance) and what remains the same (defining the continuing life of a specific person). The rejection of a computer model of brain/mind implies that it would be totally inadequate to think of the soul as the superprogramme running on the hardware of the body, but that impoverished analogy would at least point in helpful direction, however much it might require augmentation in order to correspond to the profound complexity of human nature.
As with all attempts to describe humanity, one soon reaches the hand-waving stage of the discussion. Yet the need simultaneously to acknowledge both psychosomatic unity and also the existence of a carrier of human identity is an insight that has had a long philosophical history. Aristotle spoke of the soul as the 'form' of the body; in other words, he too thought of the soul as pattern. This way of thinking was taken up by Thomas Aquinas, who rejected the Platonic dualism that had dominated Western Christian thinking since Augustine.