A paradox is a situation which seems self-contradictory or impossible, but is really true.
The most obvious paradox is where Santiago says "Fishing kills me as it keeps me alive." Fishing was Santiago's liveliehood. He had no other way to earn money, so in that sense it kept him alive (although in poverty). But, fishing also killed him. He had led a hard life. He had gone more than 100 days days without catching a fish. Therefore every day he went out he was losing life. This paradox can also be taken as a metaphor. While fishing took a lot out of him, he also loved it so it gave him a reason to keep on living.
"A man can be destroyed but not defeated" is a major theme in Old Man of the Sea (and Hemmingway's works in general). Santiago says this as he returns, utterly destroyed but, not defeated. The sharks essentially stole from him his geatest glory, they destroyed him as a fisherman (for the moment, emotionally and physically). But Santiago was never defeated. He kept moving forward. He fought the sharks as long as he could.
He then returned home with nothing but the bones of the marlin and when he could have just given up, he instead carried the mast of his boat to his hut and followed his routines for taking care of his gear. In his mind he would go fish again. Ironically, it is through this paradox that Santiago gains the respect of the other fishermen.