Before you can begin to write your essay, you will, of course, have to have a good idea of what the conflict of the story "Lady or the Tiger" is. On the surface, the story is about the conflict of the man vs. nature...that is the most powerful of the images (what could be worse than seeing the man ripped to pieces by a murderous tiger?) But that's sort of a "side conflict."
The real conflict, the heart of the story, is the conflict within the princess (woman vs. herself.) Which has she chosen? Notice the way that the author makes a point of saying that she and her dad are "semi-barbarians." This is an important point. It goes to say that no matter how civilized they dress or act (in some situations,) at their hearts is a raw, uncivilized streak. That is the nature the woman is at war with. Her civilized, "higher" nature should have her telling the man to open the door with the woman behind it. After all, if you love someone, how could you send them to their bloody death?
Of course, this is in contrast with her "lower" nature, the part that is a remnant of her fathers barbaric past. She hates the woman who is behind the door and can't stand the idea of "her" lover being married to that witch. It's sort of a case of "if I can't have him, no-one can."
The answer is not clear. If one of her "natures" was far more powerful than the other than the princess wouldn't have made her decision: "after days and nights of anguished deliberation." Her two sides were warring against each other, showing how undecided she was.
That, in a nutshell, is the conflict of this story. Good luck with your essay!