That's easier said than done! But I can try.. I'll start with what seems to be the more straight-forward concept; the soul. The 'soul' as defined from an analytical perspective is the animal in man. It is all those unconscious, and to some extent undefinable, elements of ourselves which underlay our particular life-flow and impetus. The basic instincts are one aspect of the soul, but so are more ambiguous forms such as the archetypes and psychological processes metaphorically described in myths/fairytales.
I'm still in the process of studying Jung's concept of the spirit (and defining observations of it myself), but from what I gather, it is the net result of our conscious interpretation of the content of the soul. Religion is something that is generated via the spiritual function, it is an attempt to quantify and organize into a system the ambiguous impressions of the soul. Imagination in general is the action of the spirit, this differs from dreaming as dreams are a direct experience of unconscious content. Being the mechanism of imagination, the spiritual function is what allows us humans to project into the future, to innovate and create. Spirit is what lends us the notion toward advancement, expansion and enlightenment. It is intimately related to our cognitive functions, being that it's a sort of bridge between the soul and conscious mind. The way we register visual images, especially how they are immediately interpreted by our perception, heavily involves the spiritual function. This, I believe, is a major aspect of intuition (both the intuitive functions and the greater phenomenon called as such). Introverted Judgement, both Ti and Fi, have a lot of qualities of the spiritual function as well. The theoretical, idealistic concepts generated by subjective judgement have a lot to do with imagination and the impetus toward advancement/higher goals.
In the human experience, the soul is often personified into the archetypes, especially in the case of the Anima. Jung describes the spirit as being personified in this way as well, but often in a less literal/corporeal form. From my experience/research, I believe the Animus to be largely a personification of spirit, although to my knowledge this isn't part of mainstream analytical theory.
Some quotes from Jung on the spirit:
An archetype and a functional complex, often personified and experienced as enlivening, analogous to what the archaic mind felt to be an invisible, breath-like "presence."
Spirit, like God, denotes an object of psychic experience which cannot be proved to exist in the external world and cannot be understood rationally. This is its meaning if we use the word "spirit" in its best sense.["Spirit and Life," CW 8, par. 626.]
The archetype of spirit in the shape of a man, hobgoblin, or animal always appears in a situation where insight, understanding, good advice, determination, planning, etc., are needed but cannot be mustered on one’s own resources. The archetype compensates this state of spiritual deficiency by contents designed to fill the gap.["The Phenomenology of the Spirit in Fairytales," CW 9i, par. 398.]