The movies tell you that the Witch-king is another just Nazgûl, albeit their king, but the books seemed to suggest that he was a man. And the fact that he was the Necromancer, briefly mentioned in The Hobbit, just serves to further cloud things. My question is, who and what is the Witch-king?
First off, the Necromancer of Dol Guldur, that you recall mentioned in The Hobbit, was not the Witch-king of Angmar, but his boss - Sauron himself.
Second, just like all Nazgûl, the Witch-king was a human. They were all kings of old who were given the 9 Rings of Power, and succumbed to them.
Tolkien Gateway has a very good article on the Witch-king.
... Sauron ... gave nine of them [Rings] to Mannish kings, sorcerers, and other warriors. [The Silmarillion].
With these rings the Nine achieved glory and grew wealthy, with the ability to turn invisible at will, not aging, but hating life as they were slowly drawn under Sauron’s dominion. Eventually all became Ringwraiths, the Nazgûl.
The one later known as the Witch-king was one of these, likely one of the unnamed three lords of Númenor to take the rings [The Silmarillion]
He first appeared in the histories as a Ringwraith in S.A. 2251. Being the most powerful of the Nazgûl, he became their chief, the most dreaded servant of his master Sauron.
- The Silmarillion, "Of the Rings of Power and the Third Age", p. 289
- The Silmarillion, "Akallabêth: The Downfall of Númenor", p. 267, Ballantine Books p. 320
All nine of the Nazgûl were at first men, powerful ones in fact, they were each individually corrupted by the rings Sauron gave them.
Those who used the Nine Rings became mighty in their day, kings, sorcerers, and warriors of old. They obtained glory and great wealth, yet it turned to their undoing.
-The Silmarillion, pg 346
Some texts say that three of them were great Lords of Númenor (the Witch-king being one of these). After becoming corrupted they each were completely under the will of Sauron. Specifically, the Witch-king was just considered the mightiest of the Nazgûl as well as the lieutenant of Sauron. He was a fantastic strategist/tactician apparently, as was shown in his Campaign against Arnor.
EDIT: Oh, also, the Necromancer was in fact Sauron as DVK already mentioned.