Irony is a literary device used by Edith Wharton all over the novel "The Age of Innocence", beginning with the title of the story. There is no such thing as innocence in such a time in New York, where the aristocratic families determined the fate of the city, and where the powerful mixed with the powerful to preserve the division of classes. This is in no way the description of an innocent society.
Another irony is the character of Ellen, who represents everything society condemns: She is a woman whose character is passionate, who has left her husband, mingles with men, breaks every rule of engagement, and displays a passionate and flirtatious behavior.
May, on the other hand, is virginal, innocent, soft-spoken, and childlike. She, however, is praised for these superficial qualities while Ellen is condemned when she is in fact a much learned and deep character than May could ever be.
Meanwhile, Newland is completely taken by Ellen's presence while making himself believe that who he loves is May. He cannot abandon the ways of society and yet he mingles with Ellen for reasons he cannot even understand.
This is where the irony comes in: Society lives by ridiculous standards that are to be followed with no question, yet, where is the actual desire of our heart? Can that be quantified with money or sanctified by a family name? This is unimportant to the characters in the story, and is what makes their so-called culture and education so ironic altogether.