the above answers are correct, I wish to offer another point-of-view. Perhaps it is not the conflict within the story which is most poignant, but instead the conflict created within the reader. I know that when my students read the story, a lot of conflict (external-man v. man) erupts. The students argue about what comes out of the door and why.

I think that this is the conflict which is the most important. Stockton has been criticized for leaving the story as he did. Readers want closure- Stockton does not provide closure (internal conflict for the reader and critic).

The king has no conflict at all in this story. His rules are clear, as are the punishments. He is semi-barbaric and has no personal attachments to those who have been accused. It could be said that his method of "justice" sets up a conflict, but the king himself is not plagued with any real feelings about anything. The princess's lover also experiences no conflict in this story. He does not worry about overstepping and loving someone he should not, and when he is a prisoner he trusts implicitly in the princess to determine his fate. So the only conflicted character is the princess. She is not conflicted about loving someone from a lower social station; and she suffers no conflict when she discovers the secret of the doors, obviously breaking the law in doing so. She is, however, VERY conflicted regarding which door she wants her lover to open. The conflict is between two stong emotions in this semi-barbaric princess: love and jealousy.

One of the major conflicts is within the princess.  She has to decide for herself whether she wants to tell her lover to choose the door where her lover would be killed or the one where he would end up with a woman she hated.  It's a tough decision for her -- does she love the man so much that she would prefer to have him live even though it meant he would be with a woman she hated?  Or is she selfish enough to see him die instead?  It's a conflict within herself.

If you've read this story, "The Lady or the Tiger?," and understand what each of these types of conflict is, this is not a difficult question.  Think about each character and ask yourself if he/she has any inner conflict or if something is in conflict with him/her.  For example, the King appears to have no conflict of any kind.  He says it, it happens.  He has no hesitation in sentencing his daughter's lover to a possible death, an act he justifies by explaining it away as fate or chance.  His daughter, however, experiences all kinds of conflict.  She has disobeyed the laws of the land, apparently, by loving this young man.  That's an external conflict.  She has to really work to find the secret of the doors--another external conflict.  For the Princess, though, the conflict is primarily internal: should she let the man she loves marry a woman she hates, or should she let him die so no one else can have him.  That is her dilemma, which I'm confident you can explore on your own.  As for the courtier, he has clearly broken a law (or at least offended the King), which is an external conflict.  He doesn't appear to have much conflict when it comes to choosing a door, however--something else you can explore as a potential internal conflict.  Any conflict with the tiger is obvious, so I'll let you do that one on your own.  This should be enough to get you started.

It is unclear what part of the story lines 8-10 refer to, but I will consider both lines 8-10 of the story as well as the first introduction of the accused later in the story.

"The Lady, or the Tiger?" opens with a description of the "semi-barbaric king" and his ritual of deciding the guilt or innocence of his subjects who've been accused of crimes. In the PDF version of the story I am reading, the king is described in lines 8-10:

but, whenever there was a little hitch, and some of his orbs got out of their orbits, he was blander and more genial still, for nothing pleased him so much as to make the crooked straight and crush down uneven places.

This section describes the king's demeanor and his desire to "make the crooked straight," which could refer to his sense of justice. This section of the story indicates that the king's main conflict is within himself (person vs. self) as he reconciles the different aspects of his "semi-barbaric" personality. However, the accused is not mentioned at this point of the story.

Later, a general "accused" is introduced into the story when the narrator describes the tradition of justice the king has continued. Basically, the person accused of the crime is made to enter an ampitheatre/arena and choose one of two doors. The narrator explains,

It was the duty and the privilege of the person on trial to walk directly to these doors and open one of them. He could open either door he pleased; he was subject to no guidance or influence but that of the aforementioned impartial and incorruptible chance.

Here, the narrator indicates that the ritual presents a conflict of person vs. self or person vs. tradition. The idea that the accused has "no guidance or influence" takes the society and the king himself out of the equation. The ritual is set up so that the accused imagines he can control his own fate. The two doors will reveal either a tiger, which will eat him, or a lady, who will marry him. Although the accused picks the door, the tradition dictates his options; thus, both conflicts are clearly seen here.

The particular accused of this story is the princess's lover, so the king is more personally invested in this case than he would be in a general trial. Of course the princess is invested, as well. The story never actually reveals the outcome of the trial, but in terms of the conflict presented, there is more of a person vs. person conflict here since the king has more interest in what happens to the accused. There is also a person vs. person conflict between the accused and his lover, since she gives him a hint to what door to choose (she knows which has the tiger and which the lady); however, we know she is jealous of the lady who was chosen, so we are left unsure which option she points him to. The person vs. tradition conflict persists, as well, since this is still the same ritual that is usually performed. The person vs. self conflict is less pronounced here since the accused looks to the princess for guidance.