He says that he can't be killed by a man, and then Eowyn says "I am no man" right before killing him.
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.