In the context of "The Story of an Hour," a simple definition of irony is the difference between what everyone thinks is going on and what is actually going on--in this case, the difference between what Mrs. Mallard's friends and relatives think her reaction to her husband's death is and what is really taking place in Mrs. Mallard.
For example, when Mrs. Mallard learns of her husband's death in the train accident, she goes through what everyone observing her would expect--great anxiety, sorrow, fear. The news is complicated by Mrs. Mallard's particular health problem, a weak heart, and everyone around her is trying to protect her.
The irony begins to grow after Mrs. Mallard goes upstairs, and after thinking about the fact that she is now going to be able to live her life independent of her husband, she fairly bursts with joy. Admittedly, she tries to keep this joy in check, but as she hears the birds singing, and sees Spring bursting out everywhere, she realizes that her life is now unencumbered by a relationship that repressed her. She admits, in fact, that although she loved her husband, she loved him only "sometime." Clearly, this is a woman conscious of the repression of marriage and the freedom that has just been offered to her by her husband's death.
When she goes back downstairs, every observer wants to make sure her weak heart is protected, but when Mr. Mallard walks through the door, Mrs. Mallard has a heart attack and dies, an event that's attributed by her friends and sister essentially as too much joy for her heart to bear.
The irony here, of course, is that Mrs. Mallard drops dead not from joy at seeing her husband but because she realizes that her new-found freedom is gone. Given the conventional attitudes of her time, Mrs. Mallard's death would never be attributed to its real cause--and that is the great irony of the story.