The pale young gentleman is Herbert Pocket and he explains that his father, Matthew Pocket, is Miss Havisham's cousin. Herbert was in Miss Havisham's greenhouse that day after Miss Havisham sent for him to see if he might be a suitable betrothed for Estella (she'd decided not). Herbert criticizes Estella for being cruel and haughty, and explains that she is Miss Havisham's adopted daughter raised to "wreak revenge on all the male sex."
Herbert's description of Estella sheds light on Miss Havisham's fawning over Estella and taunting Pip during Pip's visits to Satis House.
Pip is immediately struck by Herbert's open, kind personality. He explains his background and asks Herbert to correct his country manners. When Herbert asks Pip's name, Pip says it is "Phillip." Herbert, disliking the name, decides to call Pip "Handel" after Handel's piece "The Harmonious Blacksmith."
Pip is anxious to acquire genteel manners. Telling Herbert his name is "Phillip" violates the terms of his patronage, which stipulated he keep calling himself "Pip." Pip is trying to escape his lower-class past and himself.
At dinner, Herbert tells Miss Havisham's story. Miss Havisham was the spoiled daughter of a wealthy, genteel brewer. Her mother died early and her father secretly married and had a son with his cook. Though raised as part of the family, this son was rebellious and his father left him a much smaller inheritance than he left Miss Havisham, which built resentment between her and her half-brother. Miss Havisham fell in love with and was engaged to a man to whom she gave a great deal of money, and who convinced her to buy her half-brother out of his share in the brewery. On their wedding day, this man never showed up. Rumor was that he'd conspired with Miss Havisham's vengeful half-brother, though Herbert does not know the two men's whereabouts any longer. Miss Havisham was devastated and, within the house, essentially stopped time to the minute she had been betrayed.
Miss Havisham's tragedy is set in motion by class conflict: her younger brother resents her higher status as a child born to an upper class marriage. He doesn't enjoy the full range of class privileges she does because he is born out of an affair with a cook. When his father leaves Miss Havisham more money in his will, the brother seeks revenge by partnering with Miss Havisham's fiancée to ruin her forever.
All of Miss Havisham's relations were poor and all of them except for Matthew Pocket were jealous and "scheming." He alone had warned Miss Havisham about her fiancée, perceiving that the man was only superficially genteel. Miss Havisham, though, was offended by the warning and accused Matthew Pocket of trying to get her money for himself.
The distinction Matthew Pocket makes between gentlemanly manners and a noble heart is crucial and will factor significantly in Pip's growth as a character.
Herbert concludes by telling Pip that he has revealed everything he knows about Miss Havisham. He promises that nothing shall come between he and Pip in the future and swears never to inquire about Pip's patron. Pip thinks that Herbert is implicitly acknowledging that Miss Havisham is Pip's patron.
Herbert's openness and respect are further evidence of his integrity and generosity.
Herbert enthusiastically describes his own ambition of becoming "a capitalist—an insurer of ships," though he is currently working unpaid in a counting house. Pip privately suspects that Herbert will never succeed in business.
Herbert's own ambitions are in accord with the new capitalist economy of post-Industrial Revolution England.
The next day, Herbert takes Pip to Matthew Pocket's house in the countryside outside London. There, they meet Mrs. Pocket reading in the garden, blissfully oblivious to six of her children tumbling over the footstool concealed below her skirt. She and the children are attended by two frustrated maids, Millers and Flopson. Mrs. Pocket acts absent-minded and unfamiliar with her children.
Mrs. Pocket is not a very competent parent.