Round characters in literature are usually complex personalities who are realistic. They have depth in feeling and passion, as real people do. Such characters have multi-layered personalities and usually undergo some transformation. They are almost always the main characters in a text, and their transformation directly or indirectly affects the course of events in a story. Conversely, flat characters are one- or two-dimensional and lack development. They regularly play the role of supporting cast. Such characters do not possess the depth found in round characters and often depict a stereotype. Flat characters seldom affect the plot or outcome of a story.
In Chekhov's story, Dmitri Dmitritch Gurov and Anna Sergeyevna are round characters. They are the chief personalities in the story, and both come across as normal people who run the gamut of emotions, as ordinary people do. Both develop and undergo change. Dmitri's perspective of women transforms after he meets Anna. He falls in love with her. His experience is unlike any of his previous encounters with women who he, in fact, despises. His misogyny is replaced by a new respect and appreciation, especially for Anna.
Anna reevaluates her life, and she seeks solace and happiness in Dmitri's arms. Her fear is replaced by her desire to be with him. She initially questions the morality of both her own and Dmitri's actions but later freely comes to visit him in Moscow. At the end of the story, both characters have come to accept that they are in the thick of it but accept the outcomes of their actions, as the final sentence reveals:
And it seemed as though in a little while the solution would be found, and then a new and splendid life would begin; and it was clear to both of them that they had still a long, long road before them, and that the most complicated and difficult part of it was only just beginning.
Dmitri's wife, his children, and Anna's husband are all flat characters. Their only purpose in the story is to provide more detail about the two main characters. The brief references to them indicate their roles as supporting figures. The narrative does not reveal much about their personalities except for describing their attributes and how these affect the lead characters. None of them develop or transform throughout the story.
The difference between flat and round characters lies in the view the reader gets of the character. Did you only see one side of the character, one personality? Then that character would be a flat character. If, instead, the author gave you various insights to the different personalities within a character, that character would be round. In "The lady with the pet dog" both Gurov and Anna are round characters, Gurov because the author follows him and his thoughts as his opinion changes on women and love, and Anna because we see both her desire to be true to her husband as well as her desire for Gurov. THe flat characters are the supporting roles in the cast, i.e. Anna's husband, Grurov's wife.