The social significance of "The lady with the Pet Dog" relates to isolation and marital failure and infidelity. All three were growingly popular themes among Russian writers of the realist school. This indicated a growing recognition, perhaps even acceptance, of these problems in Russian society.
Tolstoy had earlier created one of the most enduring characters in Literature in Anna Karenina whose story was centered on marital failure and adultery. The social significance of Gurov and Anna is the demonstration that Russian society was changing right along with other European countries as a result of the Enlightenment, developing sciences and philosophies--like Darwinism and Rousseau's philosophical concept of the noble savage--and the earlier upsurge of Romanticism.
Gurov and Anna also indirectly point out the socially significant disparity between the punishment meted out to adulterous men and adulterous women. According to historians and sociologists, women lost their home and children (children legally belonged to the fathers/husbands), were left with no means of earning a living, were left friendless and completely ostracized from all but the lowest parts of society. Men, on the other hand, suffered no legal or social punishments of any significance. Thus when Gurov decides the affair will continue, he is concerned about Anna's protection above his own: she must come to Moscow to him so that she is less likely to be caught in their adulterous trysts.
And Anna Sergeyevna began coming to see him in Moscow. ... telling her husband that she was going to consult a doctor about an internal complaint—and her husband believed her, and did not believe her. ... Gurov went to see her, and no one in Moscow knew of it.
Gurov and Anna are both isolated. To make this point, Chekhov leaves Anna unnamed for the first part of the story: she is only "the lady with the pet dog." Gurov's isolation is pointed out since he is a married man alone at a holiday resort for many weeks without wife or children. The social significance of isolation is dramatized when Gurov becomes exasperated with his friends and acquaintances who are incapable of listening, caring or empathizing with his epiphanies of love: isolation is ingrained in social customs in the Moscow Gurov is familiar with.
These words, so ordinary, for some reason moved Gurov to indignation, and struck him as degrading and unclean. What savage manners, what people! What senseless nights, what uninteresting, uneventful days! ... the continual talk always about the same thing...