The Black Star is one possible reward for completing the quest "The Black Star." It is the alternative to Azura's Star, a Daedric artifact of the Daedric Prince Azura. The Black Star counts toward the "Oblivion Walker" achievement or trophy.
The Black Star is a re-usable black soul gem. As such, it is able to store black souls, which are always of the Grand level, along with white souls. Like other soul gems, it can only hold one soul at a time. Unlike other soul gems, the Black Star is not destroyed after its contents are used - it is simply emptied and can be filled with another trapped soul. When trapping souls, any compatible soul gem in the inventory will be used before the Black Star.
While Nelacar claims the Black Star can house only black souls, it can actually hold souls of any type, like standard black soul gems (note: this is more of a developer oversight than a bug/glitch). Considering the abundance of enemies with black souls and the scarcity of enemies with Grand-level white souls, the Black Star is recommended over Azura's Star for players who want to develop the Enchanting skill at an uber-fast pace, especially when used in conjunction with soul-trapping weapons such as Mace of Molag Bal. The tradeoff is that this makes Aranea unavailable as a follower.