The Witch-king pops up in a lot of LotR questions, and it got me to wondering about his name. Are there actually witches in Middle-earth? The wizards are Maiar, and the Elves tend to deny what they do is magic - are there any magic-using humans, and are they called witches? If not, where did the Witch-king's name come from?
In Tolkien's little conceit, the characters themselves aren't speaking English, but rather speaking Westron, a language he invented that's derived from Old English, and it is only he, the narrator, that presents the words as English words, such as "Witch-king". Here's an explanation from the Wikipedia entry on Westron:
For example, Meriadoc Brandybuck's actual name is supposed to have been Kalimac Brandagamba, short Kali (meaning jolly, merry). 'Meriadoc', short 'Merry', is designed to maintain the reference to merriness contained in the original name. Likewise Peregrin Took's actual name was Razanur Tûc, short Razar (name of a small apple). 'Peregrin', short 'Pippin' contained both the actual meaning of the full name (traveller, stranger) and the reference to an apple
Therefore, there's no real point in looking for the concept contained in the English word "Witch" just because you have an Anglicized term "Witch-king".
Faramir describes "the black arts"
This is Faramir speaking to Frodo, and it's from the chapter labeled "The Window in the West" in The Two Towers
"The Men of Númenor were settled far and wide on the shores and seaward regions of the Great Lands, but for the most part they fell into evils and follies. Many became enamoured of the Darkness and the black arts; [...]
"It is not said that evil arts were ever practised in Gondor, or that the Nameless One was ever named in honour there;
Based on that quote, I would say there is (or possibly, "was") witchcraft in Middle-earth. I think there are two basic aspects worth singling out.
Some of Númenóreans practiced "the black arts," which sounds to me more or less like witchcraft.
Since this happened because they were "enamoured of the Darkness," with a capital D, I think it's reasonable to suggest that Sauron was involved or perhaps the object of their veneration or something.
Keep in mind, though, that the Númenóreans were the "High" men, meaning they are more powerful than the regular or Middle Peoples (see the same chapter as the quote above), so unless this is a reference to smoking too much longbottom leaf (ha!), they have qualities that are more powerful than most, for lack of a better word (see the material on Denethor, who strove with Sauron).
Historically, "magic" has usually been conceived as what we would call "black magic", which we associate with witches. Even in the bible, Jesus is accused by his enemies of being a "magician" rather than a "healer"; this was an important distinction in the ancient world. Healers were good guys, magicians were shady, disreputable characters with questionable motives and possibly connected to evil and demonic forces.
Tolkien referred to the historical usage of the word "magic" in a letter to his publisher, Milton Waldman, which is reprinted in The Silmarillion:
I have not used "magic" consistently, and indeed the Elven-queen Galadriel is obliged to remonstrate with the Hobbits on their confused use of the word both for the devices and operations of the Enemy, and for those of the Elves. I have not, because there is not a word for the latter (since all human stories have suffered the same confusion). But the Elves are there to demonstrate the difference. Their "magic" is Art... and its object is Art not Power, sub-creation not domination and tyrannous re-forming of Creation.
Here Tolkien draws the traditional line between "good magic", like that of the Elves, and "bad magic", like that of Sauron and his servants. He also laments the lack of an easily recognizable word for "good magic", which was always an issue in old stories.
If we move on to the source material, and look at who is accused of being a witch by various characters throughout the stories of Tolkien, we see the same dichotomy.
In The Lord of the Rings, at least two important characters are called witches: Galadriel and the lord of the Nazgûl, also known as the Witch-king of Angmar.
The Witch-king is obviously evil; he earned his name by studying black magic under Sauron's tutelage. The fact that he is called a witch is hardly surprising.
But Galadriel is not evil, so why is she called a witch - specifically, the Witch of Lórien, or the Witch of the woods? Because men don't know she isn't evil. They fear her because they don't know who or what she is, and men always fear that which they do not know or understand.
Another word for black magic, or "bad magic", is "sorcery". We see this word in LotR as well. It appears in its English form, as well as in the languages of Middle-earth: after Minas Ithil was captured by Sauron and turned into the fortress of the Nazgûl, it was renamed "Minas Morgul", which translates to "Tower of Black Sorcery". Again, this is bad magic at work.
In short, within Tolkien's universe, there don't seem to be any green-skinned women with warts on their crooked noses, wearing black pointy hats and riding broomsticks when they aren't cackling over a bubbling cauldron full of newt eyes and lizard tails. But any evil - or apparently evil - figure who uses magic for questionable purposes is often called a witch.
Whether or not you personally believe these characters to be witches is up to your own judgement.
While I (mostly) agree with the information in the answers provided already, there is a possible aspect that I feel is worth considering, so I'll add it here:
As has been said, Tolkien's passion was language, and he invented, translated, changed, revised, and otherwise manipulated it to great extent. So it should not be assumed that our current definitions and meanings of the words he used were the intended meanings he gave to them.
In this train of thought, my personal belief is that 'witch', in this case, is actually referring to something related to bewitched, bewitching, etc. So the Witch-king might refer to the "king" of all those who have been bewitched by Sauron (his chief lieutenant, in other words). Or it might refer to the Witch-king having bewitched others, like he attempted to do with Frodo (stabbing him and turning him in to a wraith), so the Witch-king might be the king of those that he has, himself, bewitched in this manner.
Personally, I suspect it's a play on both of those concepts, while at the same time referring to any other powers granted to him by the ring he wields on Sauron's behalf (though I'm certainly not well read enough in the more obscure of Tolkien's writings to be able to provide evidence outside of my own interpretation to corroborate any of these ideas, it fits so well with what Tolkien so often did, and why, that I have a hard time believing that this description isn't at least partially relevant and applicable)