What is the situational irony in "The Story of an Hour"?

The core situational irony in "The Story of an Hour" relies on the historical context of the society in which Mrs. Mallard lives. The story depicts her societal place in the late 1800s, a time when women were afforded precious few opportunities for self-fulfillment and were expected to take great satisfaction in taking care of their husbands and families.
Before she receives the news, Mrs. Mallard "only yesterday...had thought with a shudder that life might be long." She is weary of the monotony of caring for others, particularly her husband. She longs for more.
Thus, the situational irony is that she is expected to deeply grieve the loss of her husband but is instead filled with joy, and "she did not stop to ask if it were or were not a monstrous joy that held her." This (mistaken) turn of events has allowed her a freedom she longs for, and she welcomes both his death and the change.
Furthering the situational irony, Josephine comes to the door, believing that Mrs. Mallard is making herself ill with the grief of her loss. Instead, this wife who believes herself a widow is " drinking in a very elixir of life."
The situational irony comes to new light when Mrs. Mallard realizes that her husband is very much alive, and it is she who ends up dead at the story's close—not her husband. This ironic twist shows that so much can happen within the span of an unpredictable hour.

Situational irony occurs when something is expected to happen, but the opposite happens instead. Kate Chopin's short story "The Story of an Hour" has great examples of this literary device. This short story takes place within one hour, in which the protagonist, Mrs. Mallard, learns of her husband's death.
The story begins with the line "Knowing that Mrs. Mallard was afflicted with a heart trouble, great care was taken to break to her as gently as possible the news of her husband's death." When she hears this news, Mrs. Mallard does grieve and weep as expected, but then she retreats into her bedroom alone. Here, it is expected that she will continue mourning, and her sister becomes worried about her being alone. However, Mrs. Mallard is no longer sad. She feels as if she will now be free and able to live a joyous life.
Another example of situational irony occurs at the end of the story when her husband returns home alive and well. It is expected that she will react joyfully and be ecstatic that he was not killed in the train accident as suspected. However, she suffers a heart attack and dies upon seeing him. The doctor explains her death as "joy that kills." However, we know the opposite to be true.

In this story, all the neighbors and Mrs. Mallard's sister think Mrs. Mallard is overwhelmed with grief when she hears her husband has been killed in a train accident. In fact, even Mrs. Mallard doesn't at first understand her own feelings:

There was something coming to her and she was waiting for it, fearfully.

It is only after she stops to think that the realization dawns on her: she is feeling joyful and liberated about her husband's death, not grief-stricken. This is ironic, because it is not how she or anybody else around her expects her to feel. She is not living by the "distraught widow" script. She is, as she thinks, "free, free, free!"
She, in fact, relaxes, joyfully realizing she can now be her own person.
As a final twist, however, when her husband appears and she learns the reports of his death were mistaken, she dies. All of this happens within a single hour. The irony here is that people think she has died of joy over seeing her husband alive, when the opposite is true.

Situational irony occurs in a story when the outcome of an event is the opposite of what one expects. Two paired instances of situational irony create the structure of Kate Chopin's "The Story of an Hour," both hinging on Mrs. Mallard's heart trouble and her husband's life. 
The first sentence of the story introduces Mrs. Mallard's heart trouble, implying immediately that the news of her husband's death might precipitate a heart attack. Thus, the expectation is solidified beyond what readers would already anticipate: a woman who learns of the untimely death of her husband in a train accident will be overcome with grief. The irony is that Mrs. Mallard, after retreating to her room ostensibly to grieve, finds herself experiencing joy instead. She imagines the years of freedom spreading before her and focuses on her unexpected but welcome liberation, rather than on her loss. 
The second instance of irony, which caps off the story, occurs when Mr. Mallard appears in the doorway. He has not died after all. Normally, a wife would greet such news and such an appearance with joy, relief, and even euphoria. Readers would expect the wife, when she finds out that her husband is still alive and sees him standing there, to rush to him, embrace him, and possibly weep for joy. Instead, Louise drops dead of a heart attack. Her reaction is so unusual, the doctor ascribes her death to "the joy that kills," which is ironic since it was not what really killed her.
The impact of Chopin's story depends on the skillful use of situational irony; Louise reacts to the news of her husband's supposed death (and is happy) and to his appearing before her alive and well (and dies) in the opposite fashion of what we would expect.