The word "nada" in the story stands for the nothingness that the waiter who is not in a hurry fears. He understands the old man's desire to drink in a clean, well-lighted place, though the old man can just as easily drink at home. A clean, well-lighted place is a kind of defense against the nothingness, or emptiness, that plagues all humans.

Towards the end of Hemingway's story, the waiter thinks of the Lord's Prayer interspersed with the word "nada," beginning, "our nada who art in nada." The use of the word "nada" in the Lord's Prayer is another form of expressing the belief in nothingness, as the words "God" and "heaven" have been replaced with it. In other words, the waiter does not believe in God, and, without this belief, he is faced with a kind of crushing emptiness and fear that causes him insomnia. His only relief is to find temporary respite in a clean, well-lighted place like the bar. 

Spanish for nothing, nada in Hemingway's "A Clean Well-Lighted Place" represents the author's understanding of nihilism, the belief that life is without objective meaning. Without any meaning in his life, the old man has attempted suicide, but has been saved by others.  Now that he must endure life, the old man stay late at the cafe seeking light from the darkness of nothingness--nada--to which he must return. Thus, he shuns his return to the darkness because in it he is alone with his thoughts, his despair, his isolation. Because he knows that the world has no real norms, rules, or laws, it is only the light that keeps him from thinking about this nothingness.

Likewise, the older waiter recognizes the futility of a life that is essentially meaningless.  So, he tries to keep the cafe open and light for those others like him.  After the cafe closes, the waiter stops for a drink at another place because he, too, is reluctant to return to the nothingness that awaits him in the dark.  "He would lie in the bed and finally, with daylight, he would go to sleep."  Only the light makes him forget the nada.

The old man only says, "Nada," once, and he means it literally.  The rest of the "nadas" come from the narration, which seems to be filtered through the old waiter's perspective:

It was all nothing, and a man was nothing, too...Some lived in it and never felt it but he knew it was nada y pues nada y pues nada. Our nada who art in nada nada be thy name thy kingdom nada they will be nada in nada as it is in nada. Give us this nada our daily nada and nada us our nada as we nada our nadas and nada us not into nada but deliver us from nada; pues nada. Hail nothing full of nothing, nothing is with thee...

Some have said this old waiter is the voice of the old man's soul.  Maybe it is Hemingway's voice.  But...