Awakening Emblems unlock a power when used on a party member. For example, the one given to Ameli by the Charmer unlocks the power of Celestial Hammer.
Awakening Emblems are very useful for harnessing the full power of mercenaries unlocked using Warrior Emblems.
Many monsters drop Awakening Emblems, albeit very rarely, and so these are sought after to unlock powerful new powers. These are very rare and difficult to obtain, but killing many enemies is a good way to obtain one.