In Ernest Hemingway's The Old Man and the Sea, there are multiple perspectives from which to view their roles and relationships:

Teacher & Disciple:

"If you teach a man to fish, he will eat for a lifetime."  Santiago is a kind of Christ-like teacher, and Manolin is his trusty disciple.  Santiago does not preach, but leads by example.  He instills the boy with faith by fishing the right way, going for the big fish, not for money.  More imporantly, he shows the boy how to be a "fisher of men."

The Great DiMaggio & the Up-and-Coming Rookie:

Santiago is like the great DiMaggio, who suffers from bone spurs near the end of his career.  The boy, Manolin, is the future of the franchise, a Mickey Mantle, if you will.  Like DiMaggio, Santiago must transition toward retirement, but not without a hot streak first.

Eiron & Sub-Eiron:

Santiago is the archetypal eriron (a self-deprecator who sees himself as less than he is), while Manolin is his foil, or sub-eiron (realistic ploy who functions to break mood and direct plot back to reality).

Dynamic Characters:

Santiago learns to show courage and humility in his old age, and he suffers with aplomb.  He shows the boy how to handle defeat with grace.  So says Enotes:

Manolin undergoes an important change between the beginning and end of the story. At the beginning he still defers to the wishes of his parents that he not accompany Santiago fishing since the old man’s luck has turned bad. By the end of the story, however, Manolin has resolved to go with the old man, lucky or not, in spite of his parents’ wishes.