To the narrator of "Araby," Mangan's sister represents romance and beauty. One might even call her his ideal of beauty, since he contemplates every aspect of her appearance and movement with a religious devotion. The power she has over him is a mystery, particularly to the narrator himself, since he says:

Her name sprang to my lips at moments in strange prayers and praises which I myself did not understand.

This name is so sacred to the boy that he never entrusts it to the reader. No actual name could convey the poetry it contains for him. Plato writes in the Symposium that love begins in sensuality and desire, becoming more refined as one acquires wisdom. Joyce, in sharp contrast, depicts young, foolish love like the narrator's as the purest of all.

It is appropriate that when the narrator ends the story in disillusion, filled with "anguish and anger," Mangan's sister herself is not directly involved. The idolatry he felt was never anything to do with the girl herself, whom he did not know, and to whom he could not bring himself to speak. There is no logical connection between his disappointment at the bazaar and his idealized love, but the emotional connection is clear. The feeling of reverent adoration with which he regards Mangan's sister is too pure to survive any entanglement with reality.

To the unnamed boy narrator of Joyce's “Araby,” Mangan's sister represents escape, escape from a shabby-genteel existence in which nothing much ever happens and which appears to hold out no prospects for the future. This is a life without color, without hope or excitement. If anyone needed a touch of the exotic in their life to liven things up, it's this young man.

When Mangan's sister comes along, the lad is well and truly smitten. He doesn't just have a boyish crush on her; he sees in her a chance of escape, an opportunity to enter into a fantasy life that's about as far removed from his daily humdrum existence as it's possible to get. It's easy for us to look at the boy's situation and say that it's nothing more than an infatuation, the kind that most of us develop at some point in our lives. But for the boy, caught up in the maelstrom of emotion that he simply doesn't understand, it's a deadly serious matter.

The boy is so infatuated with Mangan's sister that he resolves to buy her a gift at the Araby bazaar. It too represents an escape from the humdrum to the exotic. But like the boy's feelings for Mangan's sister, it turns out to be nothing more than a fantasy. It's important to note in this regard that neither the narrator nor Mangan's sister is ever named in the story. This highlights the utterly fantastical nature of their relationship.

Please excuse typographical error of the name Mangan.

Please excuse typographical error of the name Mengan.

As a Modernist, James Joyce has written stories in which the characters are spiritually and psychologically floundering; "Araby" is such a story.  Its narrative relates the adolescent infatuation of a young man, the narrator, and the object of this infatuation, the sister of his friend, Mangan.  Mangan's sister, whose name is "like a summons to all my foolish blood," the narrator remarks, represents the romantic and spiritual confusion and illusion of this adolescent.

In his infatuation, the narrator watches her "shadow peer up and down the street"; he lies on the floor of his parlor and watches for her to come out her door.  Her image, much like the Virgin Mary, is with him when he goes to market with his aunt, and he images that he carries the holy grail rather than a box of groceries.  Romantically, he describes his eyes as tearful, his heart floods with emotion, his body is

like a harp and her words and gestures were like fingers running upon the wires.

In short, Mangan's sister represents an idealization that offers escape from his brown existence on North Richmond Street.  In fact, he attaches an exotic nature to his infatuation as he invites Megan's sister to the bazaar.  However, like his other illusions, the bazaar is but a petty place where the peddlers engage in idle gossip.  It is then that the narrator realizes his illusions and that he has been "derided by vanity" as his eyes burn with "anguish and anger."

In James Joyce's story "Araby," how does Mangan's sister represent Ireland?

Before talking specifically about Mangan's sister in James Joyce's "Araby," it's worth mentioning that Irish literature has a long history of using female characters as symbols and personifications of Ireland. One of the most famous examples of this trope can be found in W.B. Yeats and Lady Gregory's play, Cathleen ni Houlihan. In this play, the main character (the eponymous Cathleen) begins as an old woman wandering the countryside and lamenting the loss of her four green fields. This character is often read as a symbol of occupied Ireland, as Cathleen's four green fields roughly correspond to Ireland's traditional four provinces. Furthermore, Cathleen's transformation at the end of a play into a young woman can be seen as a symbol of Ireland's projected rebirth, one that occurs once she regains her sovereignty from Great Britain.

Let's consider this trope in conjunction with "Araby." In Cathleen ni Houlihan, the character Michael Gillane becomes infatuated with the old woman (who represents Ireland), and this infatuation drives him to join a band of Irish rebels fighting for freedom. In "Araby," the main character is similarly obsessed with Mangan's sister, and the thought of her is enough to rouse him to some pretty dramatic emotion: "her name was like a summons to all my foolish blood" (30). Like Michael, the unnamed narrator of "Araby" is infatuated with a girl, and, also like Michael, this infatuation drives the narrator to perform deeds to win over said girl. The difference is that, instead of heading off to war, the narrator goes to Araby in an attempt to buy Mangan's sister a trinket. Through these parallels with the classic personification of Ireland as a female character, we can see Mangan's sister as potentially symbolic of the island itself. 

If we take this symbolism to be the case, then the end of the short story becomes very interesting indeed. At the end, the narrator fails to buy anything at the bazaar, and he realizes the foolishness of his actions and obsession. As such, it would appear that Joyce is throwing an element of disillusionment over Ireland's classic symbolic form. Perhaps, Joyce seems to be saying, it's not wise to allow metaphorical infatuation to govern our lives and drive us to perform deeds to prove our love/patriotism. In this sense, Joyce takes a step toward dismantling the symbolic female as Ireland trope.    

Why is Mangan's sister viewed as a symbol of Ireland in "Araby"?

It may be a matter of opinion that Mangan's sister represents Ireland, but knowing Joyce, it seems quite safe to make this interpretation.  The personification of Ireland is a persistent theme in Joyce's works.  Notice that the boy has never really known Mangan's sister.  This alone is worthy symbolism of an idealized Ireland, the only Ireland that a person of the boy's generation can know.  Driving the plot of the story is a seemingly hopeless quest that the boy undertakes on behalf of this woman, or his native country.  It is probably not too cynical to take the drunken uncle to represent the Irish people.  And the bazaar, England herself.  Every conceivable obstacle that can slow our boy's quest, impedes him, but he perseveres, only to reach his destination to a conclusion which seems utterly heartbreaking somehow if we take it in the context of the story.

If we accept the premise that Mangan's sister is Ireland, it is easy to consider the story to be at least partly allegorical.  For a different view of this theme, the personification of Ireland, one that almost certainly inspired Joyce, read Cathleen ni Houlihan by W.B. Yeats.