The Soul Essence Gem is an item that is used to hold part of the Dragonborn's soul if the option to become partially soul-trapped is chosen at the end of the quest "Chasing Echoes."
When found during the quest "Beyond Death," it will remove the Weakened Soul ability that is gained upon becoming soul trapped.
After retrieving the Elder Scroll, Valerica will give the Dragonborn the objective to find the Soul Essence Gem.
It can be found in the Soul Cairn within a large castle-like structure. Once picked up, it changes into a black soul gem.