An author uses direct characterization to tell his/her readers what a certain character is like. For example, an author can state that the hero in his novel is fearless, stubborn, or even meticulous. In direct characterization, the author directly reveals to us the personality of the character by using adjectives and other descriptive words and phrases.

For example, we are told that the king in the story is "semi-barbaric" or "barbaric" in nature. He is also said to be a man of "exuberant fancy," with "an authority so irresistible that, at his will, he turned his varied fancies into facts." He is introspective, a man "greatly given to self-communing."

Similarly, his daughter is directly characterized as having the same barbaric nature; she is also "intense and fervid" (passionate) in temperament.

In indirect characterization, an author reveals what a character is like through an account of his/her thoughts, actions, speech, facial expressions, and effect on others. In the story, the king commissions a special amphitheater to be built. According to the text, the king's arena is to be "an agent of poetic justice, in which crime was punished, or virtue rewarded, by the decrees of an impartial and incorruptible chance."

On any given day, an accused lawbreaker must choose between two doors to open in the king's arena. A beautiful maiden stands behind one door and a fearsome tiger behind the other. Here, the king's actions show that he is primarily focused on fulfilling the dictates of his "barbaric idealism." Through the king's actions, the author indirectly shows us that the king is iron-willed and imperious in nature.

In indirect characterization, authors may also reveal to us the thoughts of certain characters. In the story, we are told what the princess thinks of the beautiful maiden who stands behind one of the doors and what she thinks about her lover possibly meeting a violent end. Here are some quotes that reveal to us the princess' thoughts. They tell us much about the kind of person she is:

Often had she seen, or imagined that she had seen, this fair creature throwing glances of admiration upon the person of her lover, and sometimes she thought these glances were perceived, and even returned. 

...with all the intensity of the savage blood transmitted to her through long lines of wholly barbaric ancestors, she hated the woman who blushed and trembled behind that silent door.

How in her grievous reveries had she gnashed her teeth, and torn her hair, when she saw his start of rapturous delight as he opened the door of the lady!

Through her thoughts, we can see that the princess has a jealous and possessive nature. The author ends the story ambiguously; we are left to decide for ourselves whether the princess will succumb to the dictates of her possessive nature or whether she will yield to compassion. The beauty of indirect characterization is that it allows the reader to come to his/her own conclusions about a particular character.

What are the indirect and direct characterization of the two characters: Courtier and semi-barbaric king in The Lady, or the Tiger?

In "The Lady, or the Tiger?" there really are only three major characters.  The princess is the primary character, of course, but she and her actions are clearly influenced by the two characters you mention in your question--the king (her father) and the courtier (her lover).  Each of them is clearly characterized by both the description of the narrator (direct) as well as their actions (indirect).

The king is, indeed, semi-barbaric.  We know that because the narrator describes him this way in direct characterization:

He was a man of exuberant fancy, and, withal, of an authority so irresistible that, at his will, he turned his varied fancies into facts.... When every member of his domestic and political systems moved smoothly in its appointed course, his nature was bland and genial; 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.

Indirectly, we know he is semi-barbaric because he creates this spectacularly unjust and vicious form of "justice," because he blandly sentences the man his daughter loves to this court of justice, and because he is able to sit and watch without feeling the consequences of his actions.

The courtier is, the narrator says in direct characterization, a perfect example of a classic romantic hero; and it's not surprising the princess falls for him.

[He] was a young man of that fineness of blood and lowness of station common to the conventional heroes of romance who love royal maidens. This royal maiden was well satisfied with her lover, for he was handsome and brave to a degree unsurpassed in all this kingdom....

Indirectly, we know that he dared to love a princess and that he was not surprised to get caught and be placed in the arena for justice.  The best indicator of his character (which comes through indirect characterization) is the fact that he knew the princess would find out which door held the tiger and which held the fairest maiden in the land--and was willing to allow her to decide his fate.  That is a stunning display of trust; and, given the princess's semi-barbaric tendencies, he has to know it's possible this will be his last act on earth.  

The reason the reader truly does not know which door he will open is due to the courtier as much as the princess.  We know the war which is raging in her between despair at losing her lover and awful jealousy at the thought of losing him to another woman.  What is going though the courtier's mind is hidden from us except for one clear fact--he knows she will have the secret of the doors.  It's his decisive move to the door she indicates (again, indirect characterization) which leaves us in such suspense.