One conflict was that Lady Bracknell would not allow Gwendolyn to marry Jack. He was adopted and never knew his real parents. Because of this "problem" of his, he was denied her parents' blessing.
Another conflict was that Gwendolyn thought that Jack's real name was Ernest, but it wasn't. She seems to love him for his name only, and he fears he cannot tell her the truth.
These two conflicts work together up to the very end. Jack doesn't know his background or his biological parents, but claims to be "Jack." Once he finds the truth, he is actually related to Gwendolyn--and because of that, Lady Bracknell accepts his proposal to Gwendolyn. These two conflicts work together to get the main character what he wants (marriage to the woman he loves.)