The privileged piece of information that Algernon takes from Jack and Gwendolen’s conversation is Jack’s country address in Hertfordshire.
While Jack is proposing to Gwendolen, she asks for his address so she can write to him. Jack gives it to her:
"The Manor House, Woolton, Hertfordshire."
Algernon, overhearing this, secretly writes it down on his shirt cuff. This allows him to later travel to Jack’s country estate, pretending to be "Ernest", so he can meet Cecily.
This moment is crucial because it sets up one of the play’s biggest comedic misunderstandings, as Algernon arrives at the country house under false pretenses, leading to chaos when both he and Jack claim to be "Ernest."
