To add to the very significant phrases from Kate Chopin's "The Story of an Hour" that have already been posted, here are others:

She would have no one follow her.

This phrase about no one following her is the very first indication of the independence of Mrs. Mallard as she has wept with "sudden, wild abandonment" and then turned to climb the stairs.

The delicious breath of rain was in the air.

Louise Mallard senses things more now that she is free from the repression of her husband.  She looks to the West, the future, and delights freely the sounds of nature.

When she abandoned herself a little....

Mrs. Ballard is beginning to emerge as a person herself.

A kind intention or a cruel intention made the act seem no less a crime as she looked upon it in that brief moment of illumination.

Louise Ballard experiences her "awakening" as she realizes that she is free to be her own person, no longer repressed by her Victorian husband.

There was a feverish triumph in her eyes, and she carried herself unwittingly like a goddess of Victory.

Louise Mallard feels that she has won back her identity; she is free and independent of all that made her the wife of Brently Mallard.  The suggestion of war with these two phrases is also interesting as when Mr. Mallard does appear, Louise Mallard is fatally killed.

The first line that pops into my head right now is the very ending line, the one where the doctor proclaims that Louise Mallard had died "of joy that kills."  That line is loaded with meaning and irony.  First of all, there is the fact that the doctor, and everyone else, just assumed that Louise had been so happy to see her husband that she had died of a heart attack from the joy of it.  As we know from earlier in the story, that is not true; in fact, she was dismayed and dealing with her new-found freedom being torn away from her.  Secondly, it is interesting how her sister had been so careful to tell her the news of her husband's death, because she was worried that it would shock her.  It didn't as much as the opposite, news of her husband.  That was the shock that did in her heart.

Another sentence that is important:

There would be no powerful will bending hers in that blind persistence with which men and women believe they have a right to impose a private will upon a fellow-creature.

That line relays Chopin's, and Louise's, view of marriage, that it was someone forcing their will upon another person.  That explains Louise's reaction to her husband's death.  Another important phrase is when Louise declares, "Free!  Body and soul free!"  This reveals her joy at being released from marriage.  I also like the description of her face as having "repression and even a certain strength."  That tells us a lot about her, that she feels repressed, and is a strong woman.  One last line is the ever-important intro. to her heart problem, "Knowing that Mrs. Mallard was afflicted with a heart problem" that is at the very beginning of the story.

I hope that those help a bit; good luck!

Phrases which are important to the story and help move it forward are many in this short story, so I will select just a few and comment upon each.

1) Firstly, the opening sentence is key in setting up the central irony of the story, as it refers to the "heart trouble" that Mrs. Mallard has. This becomes crucial at the end of the story.

2) What is most interesting about the story is the way that when Mrs. Mallard locks herself away in her room and looks out of the window, what she sees is representative not of death but of new life:

She could see in the open square before her house the tops of trees that were all aquiver with the new spring life. The deliciouis breath of rain was in the air.

This is the first indication that Mrs. Mallard views the death of her husband not as a tragedy, but as something that bestows the gift of life to her. 

3) What then becomes important is her own realisation of her state as a widow, as she says to herself, "Free, free, free!" The accompanying physical feeling of release reinforces this speech.

4) The reference then to her own feelings of freedom and how marriage is described as having a "powerful will bending hers" captures the theme of marriage as something negative in the short story.

5) Finally, and most tragically, the short story ends with the shock of her husband's reappearance, and the dramatic irony that she died apparently of "joy that kills" because of her heart disease, whereas the reader knows it was not joy at all. 

These are some phrases that advance the central theme of the story, that of the oppressive position women were forced to accept in nineteenth-century marriages.

"She was young, with a fair, calm face, whose lines bespoke repression..."

The author does not spend a lot of time describing Mrs. Mallard's physical appearance. It is significant that "repression" is one of the few words she chooses to describe her.

"She was beginning to recognize this thing that was approaching to possess her, and she was striving to beat it back with her will..."

The realization of her situation and her longing for freedom are taboo during her time, and Mrs. Mallard does her utmost to prevent herself from acknowledging these inclinations in herself, knowing they can lead only to destruction.

"She would live for herself. There would be no powerful will bending hers..."

Women are subjugated by the will of their husbands. What they want is not important; it is what the man wants that takes precedence during these times. With the removal of her husband, Mrs. Mallard will be able to experience a freedom which has long been denied her.

"What could love, the unsolved mystery, count for in face of this possession of self-assertion which she suddenly recognized as the strongest impulse of her being!"

Self-realization is more important even than love by Mrs. Mallard; it is all-consuming.

"She breathed a quick prayer that life might be long. It was only yesterday she had thought with a shudder that life might be long."

Mrs. Mallard's married life was so odious to her that she did not feel it was worth living. This was true even though her husband was a kind man; the impulse towards self-realization inside herself was that strong.

"...she had died of heart disease - of joy that kills."

This statement has multiple meanings. "Heart disease" refers to Mrs. Mallard's physical ailment as well as to her longing. The "joy that kills" is the sense of identity and freedom she craves; in the society and times she lives in, it is a forbidden thing that can only result in ruin.

This story, although extremely short says a great deal about life (particularly life for women)  in the time period in which Kate Chopin wrote it.    Five of the most significant phrases are as follows:

“She did not hear the story as many women have heard the same, with a paralyzed inability to accept its significance.”

“She said it over and over under her breath: ‘ free, free, free!’”

“But  she saw beyond that bitter moment a long procession of years to come that would belong to her absolutely.  And she opened and spread her arms out to them in welcome.”

“And yet she had loved him – sometimes.  Often she had not.  What did it matter!”

“’Go away.  I am not making myself ill.’  No; she was drinking in the very elixir of life through the open window.”