There are two striking aspects of this dialogue. The first is its strong similarities to Plato's Phaedo (the dialogue which describes the soul, at the end of which Socrates drinks his poison Hemlock, per the ruling of execution by the Athenian court. The second is that the Christian position is being argued not by Gregory, but by one of his tutors, his older sister, Macrina. This is amazing because the conventional wisdom is that women were viewed as being incapable of theological thought. This demonstrates that in 4th century Asia Minor, this was not the case. A sidelight is that this dialogue was written shortly after Gregory's brother Basil died, and there is a suspicion that Macrina was ill when this was written. Both paralleled Socrates' death in the Phaedo.
As the title states, the main subject is the nature of the soul. Unlike the earlier reviewer, I did not find many specifics relating the soul to the four classic elements of antiquity. The primary thread which tied the soul together, as it were, is memory, kept pure of strong emotions such as lust and anger. It is memory which is how the body is reconstituted when it comes time for the resurrection of the body. Gregory even seems to ape Plato's notion of hindrances hanging onto the soul as if nailed there, to disfigure and pull down. Gregory uses the image of barnacles which attach themselves to the hull of a ship.
The Cappadocians were heavily influenced by Origin in Alexandria, but unlike Origen and Plato, the Cappadicians did not believe the soul pre-existed the body, and this dialogue makes that clear. That is why Gregory's notion of memory as a more psychological fact than as a metaphysical everlasting soul, is the heart of his argument. He even goes so far as to use the adage from the Delphic Oracle "know thyself". If one were following Plato, the emphasis would not be on knowing a personal memory, but on knowing the shared eternal memory of the ideas.