the quest is called ' The Black Star' the corrupted form of Azuras Star, travel to the Shrine of Azura to activate the quest. its called the black star and is one of the many daedric artifacts that can be obtained in skyrim.
There is a very powerful Black soul gem that can have an unlimited amount of black souls pass through it (only one at a time) without the gem disappearing. I do not recommend selling them unless you need gold or don't use the enchantment skill. Form ID: Name: Editor Name: Value: Weight: 00063B27: Azura's Star: DA01SoulGemAzurasStar: 1000: 0: 0002E500: Black Soul Gem: SoulGemBlack: 300: 1: 0002E504: Black. the souls level are much like the white souls level range(though I have yet to find a petty soul for a black soul gem) each black soul gem has a weight of 1 and a base value of 300 gold when empty and 750 gold when filled. Though Usually Grand the souls that can be trapped depends on the level of the characters soul you are trying to capture, Higher levels like bandits or other warriors obviously means higher level souls where as lower level characters like beggars give lower level souls. The purpose of this soul gem is for the capturing of black souls(Human/Humanoid characters)for use at the arcane enchanter.