In The Lord of the Rings: The Return of the King the Witch King of the Nazgul says that he can't be killed by a man, and then Eowyn says "I am no man" right before killing him. Why is he able to be killed by a woman?
In the book of The Return of the King, an appendix relates that during the Third Age the elf Glorfindel prophesied that the Witch-king of Angmar (Lord of the Nazgûl) would eventually fall, but not by "the hand of man". By the time of the Battle of Pelennor Fields, word of the prophecy had apparently reached the Witch-king himself. Like so many seemingly favorable prophecies in literature, this one turned out to be very literal and limited and thus led him to be fatally overconfident.
Source: Wikipedia article on Glorfindel.
The Witch-king actually says, 'No living man may hinder me!' to which Éowyn laughs, revealing and stating that she is a woman. There is a battle in which Éowyn cuts off the head of the WK's flying mount(his so-called 'fell beast'), but the WK has the upper hand, breaking Éowyn's shield with his mace and about to deliver the killing blow, until Merry stabs him behind his knee with his special Barrow-blade given to him by Tom Bombadil from the Barrow-downs, "forged many long years ago by Men of Westernesse: they were foes of the Dark Lord, but they were overcome by the evil king of Carn Dûm in the Land of Angmar (LoTR, "Fog on the Barrow-downs)." Éowyn then, of course, stabs him in the face with her sword. The text gives an explanation of what happened:
So passed the sword of the Barrow-downs, work of Westernesse. But glad would he have been to know its fate who wrought it slowly long ago in the North-kingdom when the Dúnedain were young, and chief among their foes was the dread realm of Angmar and its sorcerer king. No other blade, not though mightier hands had wielded it, would have dealt that foe a wound so bitter, cleaving the undead flesh, breaking the spell that knit his unseen sinews to his will.
- The Lord of the Rings, Book V, Chapter 6: "The Battle of the Pelennor Fields"
As the other answer states, this is addressed in the Appendices, which tells of how King Ëarnur, the last king before Aragorn, wanted to pursue the Witch-king but Glorfindel says:
“Do not pursue him! He will not return to this land. Far off yet is his doom, and not by the hand of man will he fall.”
- The Lord of the Rings, Appendix A: "(iv) Gondor and the Heirs of Anárion"
Ëarnur ends up going after the WK anyway and nobody finds out what happened to him, other than that he died. As for Glorfindel's prophecy, it's fulfilled by both Éowyn and Merry, neither of which is a man: Éowyn is a woman and Merry is of a different race (or species if you will). Gandalf also states this prophecy to Denethor.