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."