The conflict of this very complex story relates to allusions in the story (1) to the theme song of the actual Dublin fair, which begins "I'll sing thee songs of Araby"; (2) to Caroline Norton's poem "The Arab's Farewell to His Steed"; and (3) to the bazar being an analogy of a church.
When all these elements are taken together, it can be recognized in this cryptic and elusive story that Mangan's nameless sister is a symbol for hypocrisy and falseness in the Church (the bazar) and in religion in Ireland.
In light of this, the conflict is best described as Human (Man) versus Society. If we only focus on the narrator's actions, it is possible to ascribe a less acceptable Human versus Self conflict, but this approach leaves so many threads of the stroy unaccounted for and raises so many more questions than it answers that it must eventually be seen as an unsatisfactory analysis of the conflict.
To cover further detail, the theme song of the fair, "I'll sing thee songs of Araby," is about bright illusions, "rainbow visions rise," that "cheat thee of a sign." This confirms Mangan's sister as one who cheats the narrator by tempting him insincerely with a bright sign. The meaning of Norton's poem, which ends happily though with a note of upcoming repraisal, "who overtakes us now shall claim thee for his pains!" is about betraying that which is precious.
This poem meaning is the inverse meaning of the song, thus one allusion represents the story addressing the reader from the girl's point of view while the other allusion represents the story addressing the reader from the boy's point of view. The girl's point of view is: I shall weave illusions to attain what I want from you. The boy's point of view is: I have been betrayed and sold away for gain (with the hope of a happy ending for his fate as well).
When these allusions and their meanings are taken into account, the primary conflict is clearly Human versus Society (since Mangan has a larger role than a romantic interest as the symbol for the Church and for religion), with a secondary conflict of Human versus Human or versus Other.
I heard a voice call from one end of the gallery that the light was out.
Gazing up into the darkness I saw myself as a creature driven and derided by vanity; and my eyes burned with anguish and anger.
[Allusions taken from "Wallace Gray's Notes for James Joyce's 'Araby'."]
The main conflict in the story is character vs. self, the result of the narrator’s shyness and loneliness.
The narrator of the story is a young man who seems unable to find his way in the world. He thinks he is in love with a girl, and wants to get her a present. However, he is not able to do it. First, he can’t really talk to her. He is obsessed with her, but it is a confused obsession.
Every morning I lay on the floor in the front parlour watching her door. The blind was pulled down to within an inch of the sash so that I could not be seen. When she came out on the doorstep my heart leaped.
Second, he can’t seem to get to fair to get her the present before she leaves. He waits for his uncle to give him money, but even once he makes it to the fair he is unable to look for a gift. His motivations are mixed and confused, and he leaves unsatisfied.
Although he has issues with the girl, his uncle, and the street fair workers, the narrator's most serious issues are within himself.