Organisms derived from the LCL of Lilith require an A.T. Field (and hence a soul) in order to maintain physical integrity. A lifeform comprised of particle-wave matter is able to exist independent of the soul, whether it was created without one in the first place (e.g. the Evangelions and Rei clones) or if the soul was subsequently removed or lost (the Seeds and various dead Angels).
Souls originate from the Chamber of Guf, entering and leaving the world through the Doors of Guf. Souls then intersect with LCL, and by a process that Aoba calls "self-formation of LCL", they form a vessel that is shaped by the soul using the A.T. Field/ego border/edge of the mind/self-image. The words soul, mind, self-image, psyche, heart and will seem oftentimes interchangeable in NGE.
Episode 20 reveals the most about the soul's physical properties. A torus electromagnetic is placed around the Entry Plug to re-anchored Shinji Ikari's soul within the LCL present within the plug. Libido, Destrudo, and the temperature of the LCL also seem to be factors during the salvage of Shinji. The entire salvage of Shinji during this episode is based on the salvage attempts of Yui Ikari after the Contact Experiment, giving us one of the few windows present during the course of the show into exactly what is involved between the soul and body.
Souls appear as red orbs of light during The End of Evangelion. In the Old Testament/Torah blood (and so possibly LCL) is equated with the lowest aspect of the soul in Kabbalah called Nefesh (נפש). See Leviticus (ויקרא) xvii:xi and Deuteronomy (דברים) xii:xxiii.
As this term is so closely tied many other terms in the show the following articles are recommended reading.