In basic terms, a nihilist is someone who believes in nothing, a person for whom life has no worth or value. Such a tragic attitude is often adopted by people who've been dealt a bad hand in life or who have no friends or family or the love that comes with them.
This seems to be the unfortunate situation that the older waiter in Hemingway's “A Clean, Well-Lighted Place” finds himself in. Here is a man whose life is empty, a man without anything in life to look forward to. After he finishes work at the café late at night, there's nothing for him to do but go home and sleep.
To be sure, the waiter tries to fill the void at the heart of his meaningless existence by lingering on at the café, which, with its bright lights, is at least a clean, well-lighted place. This is more than can be said of the place where the waiter lives.
But as a nihilist, his heart isn't really in it. And besides, the café represents a very poor substitute for the consistent stability and meaning that the waiter desperately needs in his life. Without such stability, without such meaning to his daily existence, the waiter will continue to be a nihilist.
It is not surprising, then, to hear him recite a parody of the Lord's Prayer, in which he uses the word nada—Spanish for “nothing”—over and over again. This quasi-religious confession amply demonstrates the fact that a nihilistic void now exists where his soul used to be.