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Spectral Essence is taken from the remains of an Eidolon (or Ice Beast) after it's slain. According to the Seers, the Eidolon's body has a natural resistance to the effects of Spectral Agony, the devastating power that the Mursaat use to decimate their foes. As such, this Spectral Essence is sought by players to infuse their armor.
The essence seems to be an entanglement of tentacles from the Eidolon.
Note that Spectral Essence is a bundle item - meaning it is carried in your hands, and not your inventory. Eidolons always drop Spectral Essence.