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.    

All Mangan's sister does is to mention how much she would like to go the bazaar called Araby, which is coming to Dublin. She thinks it will be "splendid," but she is going away on a school trip, so she can't attend.

The narrator has a crush on Mangan's sister, who has no other name in the story, so when she speaks to him about the splendid bazaar, it gets conflated (combined) in his mind with his desire for her. His desire for her is heightened as she speaks about Araby, for the light illuminates her neck and the white hem of her petticoat. He mentions getting something for her at the bazaar:

If I go, I said, I will bring you something.

From that time on, the narrator dreams of going to the bazaar, and can't concentrate on anything else. School, schoolwork, and the neighborhood seem dull and mundane against his expectations:

The syllables of the word Araby were called to me through the silence in which my soul luxuriated and cast an Eastern enchantment over me.

However, the enchantment will prove to be an illusion.

Araby by James Joyce is one of the short stories from the Dubliners series in which Joyce explores various life stages or potentially transformative events which stand to change the lives or circumstances of the characters.  

The boy in Araby is infatuated by Mangan's sister and is apparently coming to an age where awareness of, in this case, girls, is still bewildering. Even the boy does not really understand his feelings, his "confused adoration." He has barely ever said a word to her and yet he idolizes her. Thoughts of her invade all his activities and he takes every opportunity to watch or think of her:

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

The first time she speaks to him he is overcome by the "curve of her neck" and does not remember part of their conversation, only that he is alone with her. He undertakes to being her something from the local bazaar and is then consumed by all thoughts of the forthcoming event. Only later does he accept that she will not have waited with the same anticipation. 

The boy's thoughts are typical of a young boy beginning to explore his attachment to girls and the emotional feelings that accompany his expectations. He barely knows what to expect but, when he does actually make it to the bazaar, he is overcome by his confusion and, instead of buying something, he stands dazed and somewhat perplexed by the situation. He has been so eager to go to the bazaar and to bring something back for the girl but, now that he has the opportunity, he is unable to complete the transaction which would have transferred his childish imaginings and adoration into something more concrete which maybe he is not ready for. The boy comes to a realization that perhaps his feelings were exaggerated and more about himself than the girl.  

The unnamed boy is young, innocent, naive. His deeply unhappy home life causes him to fantasize about an exotic, more exciting world in which all his deepest, most heartfelt dreams come true. The bazaar is that world—or at least it appears to be.

Sadly for the boy, Araby turns out to be every bit as much of a disappointment as his ordinary everyday life, with its seemingly endless disappointments. All he wanted to do was buy Mangan's sister a gift, something special that would show how deeply he feels towards her. But he's unable to do even that.

The darkness that descends upon the bazaar represents the end of his dreams, the onset of a profound disillusionment with a harsh adult world. For the boy, this is the end of innocence. Araby, like his infatuation for Mangan's sister, was all just an illusion.

Mangan's sister doesn't need to do anything special to make the trip to the bazaar important to the narrator. He is so enamored of her that the fact that she simply speaks to him about the bazaar would probably be enough to make him want to go. He says,

At last she spoke to me. When she addressed the first words to me I was so confused that I did not know what to answer. She asked me was I going to Araby. I forgot whether I answered yes or no.  It would be a splendid bazaar; she said she would love to go [....]. She could not go, she said, because there would be a retreat that week in her convent [....]. The light from the lamp opposite our door caught the white curve of her neck, lit up her hair that rested there and, falling, lit up the hand upon the railing.

Not only does the narrator hope to make her happy by bringing her a gift from the exotic Araby, but he describes her appearance as though she were magic, lit up, and purely beautiful. Even before she speaks to him, he thinks of himself as her champion, a hero. Just prior to their conversation, he says, "I imagined that I bore my chalice safely through a throng of foes. Her name sprang to my lips at moments in strange prayers and praises which I myself did not understand." When Mangan's sister mentions her disappointment that she cannot go to the bazaar, she, in essence, gives him his quest.

I disagree, incidentally, that the narrator's failure to name her means that his childhood feelings for her were, in fact, silly or childish. He is unnamed as well. As the other commenter states, this is a coming-of-age story, and the narrator does lose his innocent naivety, but the namelessness of both himself and his childhood love serve to make this story feel more universal.

It is clear that this short story is a "coming of age" story where the narrator is a tender adolescent boy who is completely overcome by his Romantic ideas of love. These certainly come to focus on Mangan's sister. Note how she is described from the point of view of the narrator:

She was waiting for us, her figure defined by the light from the half-opened door... Her dress swung as she moved her body and the soft rope of her hair tossed from side to side.

Note how in this description the light that surrounds her gives her an almost angelic appearance - it is as if she has a halo. This Romantic obsession the narrator has with Mangan's sister (note how she is never named - this itself seems to show the foolishness of the narrator's feelings) clearly dominates him, as he himself expresses later on in the short story:

But my body was like a harp and her words and gestures were like fingers running upon the wires.

This sets the stage for Mangan's sister's appeal to the narrator to buy something for her, clearly placing him in the role of knight errand off to complete a dangerous quest on behalf of his beautiful lady who awaits his safe return. It is this that sets the narrator up for his epiphany at the end of the story and makes him realise his own vanity.

The fence physically separates the narrator and Mangan's sister, but it also symbolizes how untouchable she is emotionally. The boy exalts and worships Mangan's sister so much that she seems almost unreal, or at least unreachable to him.

In a cultural sense, the fence also symbolizes how "cloistered" Mangan's sister is, as well as the other young women in her convent school. She mentions that she cannot go to Araby because of a school event. One gets a sense that the girls of Richmond street do not enjoy quite as much freedom as the young boys do.

The fence symbolizes the fact that the sister is unattainable.  The protagonist, realistically, will not be able to woo the sister, who is not interested in him romantically.  She is his first real crush and this is evident by his actions, like neglecting everything else in his life to watch her.  He finally realizes, though, at the end of the story, the reality of what time he has wasted obsessing over this young woman.  This reality is a sobering one for the young protagonist.  He realizes that he has much to learn about life. 

In the narrator's eyes, Mangan's sister represents an idealized romantic figure. For him, she is an intoxicating combination of beauty and hope, and his fantastical longing for her takes him out of the confines of his dull everyday existence and gives him a tantalizing glimpse of a much richer, more exciting world.

The powerful effect that Mangan's sister has on the boy can be seen in the almost religious way he describes her, as if she were the Virgin Mary herself:

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

There is clearly something about the girl's beauty that is pure and transcendent, something that cannot adequately be captured in ordinary, everyday language. The very fact that the boy cannot bring himself to utter the name of Mangan's sister is a sign of just how sacred she is to him.

Having idealized Mangan's sister, the boy sets out on a kind of Arthurian quest to buy her a gift at the Araby bazaar. His eventual failure in that quest immediately dispels the romance with which his mind has recently been so obsessed. The powerful ideal of romantic beauty that he has found so all-consuming and intoxicating vanishes, leaving the boy crushed, humiliated, and thoroughly disillusioned.

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.

It's a pretty dark and bleak world in which the narrator lives. His street is "blind" and "quiet" with a "brown imperturbable [face]." The home's former tenant, a priest, had died inside, and the garden behind the house had grown "wild." The story is set during winter, when "dusk fell before" dinnertime, and the narrator and his friends would play outside in the "dark muddy lanes" and "dark dripping gardens," near the "dark odorous stables." The repetition of the word "dark" is symbolic; this is a dark time for Ireland: its slums are some of the worst in Europe, political tensions are high, and Nationalists see the advent of WWI as an opportunity to rebel against England while she is otherwise engaged.

However, Mangan's sister is always characterized by light, in comparison to the darkness everywhere else. As she stands in the doorway, looking for her brother, the narrator sees her "figure defined by the light from the half-opened door." Later, when he speaks to her, "The light from the lamp opposite [his] door caught the white curve of her neck, lit up her hair that rested there, and, falling, lit up the hand upon the railing." Now, the repetition of light is symbolic, and she seems to represent something hopeful. Perhaps she is meant to represent the writer's hope that, someday, Ireland will not be the bleak and sad place of the narrator's youth.  Therefore, Mangan's sister is Ireland's brighter future.