The Tyrant Relic is an egg that, according to our mentor as well as other characters, will hatch into Solus, the legendary Tyrant of Light that will be able to put an end to the war between Tyrants that desolates the world.
We receive it from our mentor after completing the Orphanage in Peril quest. After this, we're directed by other characters such as Deena and Eliza to find shrines where we'll be able to feed it elemental power until it hatches.
It doesn't hatch, no matter how much elemental power we feed to it. But as we do this, we gain the ability to control different elements.