What is the nature of the conflict in this story? Who, or what, do you see as Mrs. Mallard's antagonist?

The main conflict in "The Story of an Hour" is character vs. society.  Mrs. Mallard's adversary is not her husband, as some might argue, but society, due to society's expectations for right female behavior as well as marriage.

First, Brently Mallard was a good husband.  Mrs. Mallard "knew that she would weep again when she saw the kind, tender hands folded in death; the face that had never looked save with love upon her [...]."...

The main conflict in "The Story of an Hour" is character vs. society.  Mrs. Mallard's adversary is not her husband, as some might argue, but society, due to society's expectations for right female behavior as well as marriage.

First, Brently Mallard was a good husband.  Mrs. Mallard "knew that she would weep again when she saw the kind, tender hands folded in death; the face that had never looked save with love upon her [...]."  She doesn't take issue with him, in particular, but rather the institution of marriage, in general, and what it meant for the woman.

She feels a "monstrous joy" because, from now on,

There would be no one to live for her during those coming years; she would live for herself.  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. 

The joy is somewhat monstrous because it comes at the expense of her husband's life; however, though monstrous, she does happy because she recognizes, as she says, that she can now be "'free, free, free!'"  In marriage, a woman of this period loses her identity as an individual; she becomes someone's wife, and her husband is legally entitled to make any and all decisions of importance.  Her duty is to acquiesce to his wishes.  She would likely not have married had this been a viable social option, but it was not during this era.  This wasn't the arrangement just for her marriage, but for all marriages.  Even though her husband was kind, she was still required to allow him to "live for her."  As someone's wife, she could not live for herself.  As a widow, she will now have that right.  Without the social expectations surrounding marriage and the prescriptions for her behavior within that institution, Mrs. Mallard would have had no adversary.