Papa is the emotional heart and soul of the story. He is larger than life, which is adequately described by his birth, which happens amid the explosions happening at the Bombay docks (1.22). He's generally described as a "man of large appetites" (1.35)—we're talking appetite for food, for success, and for love. His big heart overwhelms all his other qualities, and he fiercely supports and protects his family.
He's the man of the house, and he's hugely motivated by personal pride, bulldozing through life with his eyes on a fixed target. As can happen with people who put blinders on, his ambition is sometimes unwise. The best example of this is when he makes enemies with half of Mumbai and gets the family run out of town. The good news, though, is that these actions come out of his stubborn will and lack of good judgment rather than being a selfish man.
Papa's not a refined or elegant man at all; he's literally the opposite. The story gives us several choice descriptions of Papa… including having his daughter squeeze boils out of his back. This serves as a contrast to the refined French ways that Papa and his family encounter, which is embodied particularly in the character of Madame Mallory (see her write-up elsewhere in this section for more detail).
The thing about these weaknesses, though, is that Papa's vulnerability only makes him more endearing. He wears his heart on his sleeve, so to speak, and doesn't put a high price on surface appearance. Think of the scene in Chapter 5 where he chases Mallory out of the town square barking like a dog. We meant it when we said he's not one to care much about fitting in.
At the risk of oversimplifying Papa, though, it's important to note that Hassan calls Papa "a complicated man, not easily put in a box" (1.72). He's emotionally driven, but has a serious business-savvy side to him. Case in point: He takes grandfather's humble beginnings and turns it into the loud glittering Bollywood Nights, a business move that enables him to support his large family.
Papa is an important part of Hassan's success, mainly because he spends his whole life investing in his family. His famous catchphrase is "'Remember, Hassan, you are a Haji,'" something he repeats to encourage his son and remind him of the strength he possesses. Papa's warmth and support remains with Hassan even after his death, and when Hassan wins his third star he says that he feels "my father's mountainous presence at my side, beaming with pride" (19.19). Papa may be gone, but his larger-than-life personality lives on anyway.