A Clean, Well-Lighted Place
We might look at the older waiter as the stable center point of this story. The younger waiter is bouncing around, full of energy and impatience, while the old man waits quietly to pass out of this life. The older waiter is somewhere between these two stages of life, and observes it all from a certain critical distance. He recognizes parts of himself in both the younger man and the older one, and he's the one that makes us see that these different stages of life are universal and inevitable. We might identify him as a representative of the midlife crisis stage.
The older waiter is also the one to reveal the story's philosophical backbone to us: as he sees it, the best thing you can do is find a clean, well-lighted place in which to pass the time, because life is simply nothing otherwise. He discounts the importance and meaning of any unifying spiritual principle, mocking organized religion through his rewriting of the Lord's Prayer ("Our nada who art in nada, nada be thy name thy kingdom nada thy will be nada in nada as it is in nada" ). Instead of finding sanctuary in the belief in some higher power, the older waiter believes that there's nothing ("nada") out there, and that our lives have no greater purpose.
This embrace of the "nada" that makes up real life is what gives the older waiter his patience and calm – after all, in a world full of nothing, why rush to do anything? Why bother trying to get home early? As he sees it, happiness is just a temporary illusion, and eventually, all of us just get swallowed up by the "nada."