Just like pretty much every male character in this story, Paul is a source of heartache for the women in his life. He's Madame Aubain's son and Virginie's brother, and we don't know much about him as a child because he's shipped off to school pretty early:
He made his farewell bravely, pleased to be going to live somewhere where he would have friends. (2.54)
Everyone forgets about him pretty quickly, because the golden child Virginie is still at home.
Maybe being second-best takes its toll on poor Paul. He turns out to be a good-for-nothing adult:
He was unable to settle on any career, spending most of his time in taverns. [Madame Aubain] would pay his debts, and he would incur others. (3.86)
He seems pretty irresponsible by this account, and not a very good son.
These facts are confirmed when he gets married. Even though he has started working at the Registration Service, he's still a jerk. When his mother dies it takes him ten days to show up:
The daughter-in-law searched in drawers, chose some pieces of furniture, sold the others, then they went back to the Registration Service. [. . .]The next day there was a poster on the door. The apothecary shouted in Félicité's ear that the house was for sale. (4.43-45)
Paul doesn't even have the decency to tell Félicité, who helped raise him, that he's selling the house she lives in. He really doesn't have any redeeming qualities, unfortunately, and seems to be an example of the cold-hearted bourgeois men who make Félicité's life that much more miserable. Paul definitely earns the title of the guy we love to hate.