The old man is a regular in the café, and though he sometimes forgets to pay, he's generally a good customer. We mostly learn about the old man from the conversation of the two waiters; we're not sure what their relationship is to him, but the older of the two waiters seems to know quite a lot. We learn that the old man once had a wife and possibly a family, but now is alone with his niece, who saved him from a suicide attempt. We wonder what the old man was like before he was an old man – are his dependence on alcohol and his suicidal depression merely consequences of his age and solitude, or have they plagued him for his whole life?
Ultimately, though, these questions are irrelevant: the old man is something of a mystery to us, albeit one that doesn't have to be solved. Hemingway presents him as a representative of all people nearing the end of life, weary and hopeless, but still dignified. The key here is dignity – Hemingway wants us to see that even when life gets you down, you should accept it and try to keep it real.