The Odd Couple takes place in three acts, all in the New York City apartment of Oscar Madison. The play was written in the 1960's, but the message is somewhat universal, so that it could be (and often is) performed in a modern setting as well.
In the opening scene, Oscar and some friends are in the apartment playing poker. Oscar's apartment is terrifically messy, which is a running gag through the play (and even more so in the television series that followed). It is a hot summer evening, which leads to the general crankiness by the men in the scene. As the play begins, we learn that their friend Felix Unger has been missing since breaking up with his wife earlier that evening. Felix eventually comes to the apartment and they talk about his broken relationship. The poker game dissolves, and Oscar invites Felix to stay with him in the apartment.
In the next scene, some time has passed, and it is another poker game night in the same apartment. However, this time the apartment is spotlessly clean, and we see the effect that Felix has had by staying in Oscar's apartment. As the dialogue goes on, we also learn that, along with cleaning, Felix has been getting on Oscar's nerves and the friendship is becoming tense.
The final act of the play is, again, in the same apartment. The tension between Oscar and Felix reaches its peak as Felix is continuously cleaning and driving Oscar mad. We see ways that this has interfered with Oscar's dating life. Eventually, the two agree that they cannot continue living together, and Felix leaves, but there is an indication that the friendship will continue.
The setting, though simple in the single apartment, contributes to the advancement of the story as we see changes in the cleanliness of the apartment, and we learn of the impact of Felix's cleaning, both on his marriage and on his friendship with Oscar.