Exposition refers to the "set up" of information in a story. This includes the setting, the introduction of the characters, and the establishment of the conflict.
In "The Last Leaf", we learn that the story takes place in New York City, particularly Greenwich Village. Sue and Johnsy are the main characters, roommates who have come to the city from other locales with an interest in art, suggesting they are both of a creative turn of mind. Other characters include the doctor and Mr. Behrman, an old man and painter who lives on the ground floor of the building.
The story begins in November, and O. Henry personifies Pneumonia as if the condition is an unwelcome visitor to town. This is the conflict - Johnsy has pneumonia, and has little chance of surviving. The conflict is an internal one. Johnsy is convinced she will die when the last leaf of the season falls off the ivy tree. It isn't the disease that is dooming her - it is her own perception.
The rest of the story shows how true the effect of perception can be, as we see Johnsy get better from the influence of a trick being played upon her. The end of the story is an exercise in irony, as the one to play the trick - Mr. Behrman - succumbs to the illness that was trying to take Johnsy.