Black souls are NPC souls (bandits, city guards, merchants, farmers, etc.); white souls are everything else.
Normally you cannot capture a black soul unless you have a black soul gem (which are hard to find), or the Black Star. Either version of Azura's Star will function as a reusable grand soul gem, however only the Black Star will allow you to capture black souls (and, although it may be a bug, it also will capture white souls).
Regardless of level, a black soul is equivalent in power to a grand soul, making it easier to get your most powerful enchantments by taking out easy bandit camps. My personal recommendation is to get the Black Star, especially with its current behavior to also capture white souls.