Exposition refers to the background information that authors sometimes provide, most often at the beginning of a story, so that readers can understand a bit more about the characters and their situation. In this case, the information provided regarding Della's small savings, their apartment, Jim's salary, and the couple's two prized possessions (his watch and her hair), are the story's exposition.
The rising action begins when Della has stopped crying and leaves her flat. The narrator says,
On went her old brown jacket; on went her old brown hat. With a whirl of skirts and with the brilliant sparkle still in her eyes, she fluttered out the door and down the stairs to the street.
She has, evidently, had some kind of revelation as her passive weeping has stopped and she now "whirl[s]" and "flutter[s]" out the door with a "sparkle" in her eye. The action continues to rise as Della actually does sell her hair, as she frantically shops for the perfect gift for her husband, as she attempts to curl her newly cropped hair into something better than "ravages," as she waits for Jim to get home, when he does return, and as he sees her, for the first time, without
anger, nor surprise, nor disapproval, nor horror, nor any of the sentiments that she had been prepared for. He simply stared at her fixedly with that peculiar expression on his face.
The rising action continues as Della prepares dinner and as Jim pulls a gift from his pocket "and [throws] it upon the table." We still do not know if he is upset or merely surprised about Della's hair, and neither does she. This keeps the tension building up to the climax. We finally understand Jim's feelings when he opens the gift from Della and he says to her,
"Dell, . . . let's put our Christmas presents away and keep 'em a while. They're too nice to use just at present. I sold the watch to get the money to buy your combs. And now suppose you put the chops on."
In this moment, the story's climax, Jim's reticence to speak is now explained. He was never angry, just in some shock that his gift was rendered (temporarily) useless by his wife's generosity and love. The tension disappears as both Della and we, the readers, understand Jim's behavior.
The story really doesn't have any falling action. We don't learn any more of Jim and Della's evening or life their together after the climax. Instead, the narrator provides some commentary, a resolution of sorts, about what makes Jim and Della's relationship so special and why their gifts, gifts that they sacrificed to give out of their love for one another, are so valuable.