Shadow probably refers to a part of the unconscious mind that symbolizes repressed weaknesses or faults, as outlined by the Swiss psychiatrist Carl Jung (see also: Shadow (Persona)). It could also be related to a kind of ghost called the Shade.
A Shadow is first encountered in Mazurka Corridor on the 7th floor, where a man will beg Akemi Nakajima for help and ask for a weapon. Regardless of the answer given, he will transform into a Shadow and attack.
"It can transform into any person it sees. It is said that whomsoever meets it will die within a few days, and much folklore stems from this creature. They may also become physical incarnations of psychological ailments."
—Shin Megami Tensei III: Nocturne Compendium
Shadows appear as medium-leveled demons which the Demi-Fiend encounters after his transformation into a demon. It can easily be found in the Tower of Kagutsuchi. Being a Foul Race demon, the Demi-Fiend can only interact with it during the Full-Kagutsuchi phase or by having the Jive Talk conversation skill.
"A phantom that appears as the person who sees it. It is said those who see it will die within a few days. There are similar tales around the world, and some of these cases of sightings have been reported by mentally ill patients."
"A ghastly being that appears in mirror image of its victims. It is rumored that within a few days, death will come to those who see them, There is much lore in multiple cultures pertaining to this phenomenon. Some say that it is a sign of mental illness."