Breakfast at Tiffany's is set in New York during World War II. The war doesn't figure prominently in the story in that we don't see the main characters in combat and most of them seem untouched by the realities of war. But the specter of war sort of hovers throughout the novel as we hear about rationing things like peanut butter, we learn that most apartments lack of telephones during this time, and most devastatingly, we are told about the loss of Holly's brother Fred who dies while serving as a soldier.
The city of New York figures pretty prominently in the story. Holly loves the city, loves the Brooklyn Bridge, loves the "lights, the river" (12.4), and she vows to return with her kids so they can see the things she loves. New York offers an exciting background to the story and it lends the narrative a definite energy. It's hard to imagine the story of Holly Golightly happening anywhere else.
Much of the action takes place in or around an old brownstone apartment building where both Holly and the narrator live. It's here that the narrator first meets Holly, here that they develop their friendship, and here that most of the major events occur (Holly learns of Fred's death in her apartment, she gets arrested in the narrator's apartment, she's reunited with Doc in the building, and she and the narrator have their first argument there). It's like a little world with its own good times and bad times, its own conflicts and eccentric characters, and its own memories (the narrator has to move after Holly leaves because the building starts to feel "haunted" (19.1).
Holly's apartment is also its own special place. It's little and messy and noisy and it tells us a lot about her character. She doesn't bother to furnish her abode (except for the bedroom, but just a little bit), which adds to the general sense of impermanence that surrounds her (take a look at the "Themes" section in this guide for more on this). It suits Holly and it becomes an exciting destination for the narrator, who never knows who or what he'll find there.
So the setting is a mixture of all of these elements. It's the war and New York and the brownstone apartment building. And it's Holly's apartment and the things this space represents for the various characters.