Algernon tells Jack that before he can give his consent to Jack marrying Gwendolen, he must first clear up the "name question"—meaning Jack’s real name is not actually Ernest, which could be a problem since Gwendolen is obsessed with the name.
Algernon discovers Jack’s deception when he finds his cigarette case, inscribed with a message from "Cecily to her dear Uncle Jack." This forces Jack to admit that he goes by "Ernest" in London but is actually named Jack and has been leading a double life.
Since Gwendolen has already declared her absolute devotion to the name "Ernest", Algernon suggests that this could complicate Jack’s proposal. The "name question" becomes a running joke throughout the play, as both Jack and Algernon later attempt to get christened as "Ernest" to secure their engagements.
This highlights Wilde’s satire on superficiality—Gwendolen (and later Cecily) cares more about Jack’s name than his character, reinforcing the absurdity of Victorian social ideals.