In Das Kapital, Marx describes alienation as a worker's separation from the product the worker produces. In other words, the worker sells his wage labor to the capitalist (ceo, owner of the company) in order to make a wage in order to live (make a living). The worker makes a product but that product belongs to the capitalist (and eventually the consumer). The worker is alienated from the product and, in a sense, the world of making the product (the work environment and social/economic world themselves).
But in general, alienation means to feel foreign in the world or society you are living in. Given the above description, sometimes this means to feel alienated because you are a low level worker in a society where capitalists oppress workers. There is no question that the narrator feels alienated when he goes to the bazaar. He basically felt at home in his previous world, dreaming of Mangan's sister, enjoying literature, etc. But when he gets to the bazaar, he realizes it is not some exotic place ("Araby" - Arabia, conjuring visions of the Middle East). He also realizes that it is run by English sellers only out to make a profit. This shakes him out of his quest for Mangan's sister, which is economic, romantic, and religious (he viewed this as a romantic and religious quest). He now feels alienated from all of these hallowed institutions because of the commercial aspects of the bazaar. "Gazing up into the darkness I saw myself as a creature driven and derided by vanity; and my eyes burned with anguish and anger." He realizes these exotic products he was after are being sold by English business people. Supposing these products actually came from a far away land, this is Marx's definition of alienation: the product is made by workers and then owned and sold by the owning/capitalist class; so those hypothetical workers are also alienated from their products—perhaps by English colonialists. On some level, the narrator is also angered by the commercialization of the bazaar and how this superficiality mocks his romantic quest. In short, he feels out of place at the bazaar but he also now feels out of place with his previous mindset regarding love, religion, etc.
This story was published in 1914, a time when Ireland was engaged in debate and wars over establishing Irish Home Rule or remaining under British rule. This is significant for the end of the story because the merchants at the bazaar are English. Whether or not this was Joyce's intent, this being a Marxist interpretation, the narrator's recognition of the English accents is significant. Given that the English were viewed (by some) as a force of oppression in Ireland, we have a dialectic of class struggle between the English and the Irish. Dialectics is the study of how two opposites interact, change, and evolve. So, we might consider how the narrator's thinking changes when he (an Irishman) comes to grips with an English hierarchy.
But, to take the idea of dialectics more broadly, Marx and Engels described the revolution of the Proletariat in terms of dialectics. The bourgeoisie and/or capitalist elite oppress the Proletariat and this imbalance of opposites would eventually lead to the Proletariat rising against the bourgeoisie. Thus, the interaction of these two opposites leads to a change. In "The End of Classical German Philosophy," Engels writes:
For dialectical philosophy nothing is final, absolute, sacred. It reveals the transitory character of everything and in everything; nothing can endure before it except the uninterrupted process of becoming and of passing away, of endless ascendancy from the lower to the higher.
In "Araby," the narrator is coming to the bazaar with notions of his romantic, religious quest. For Marx, the ideas and thoughts of culture are formed by the material exchanges of culture: social, economic—the stuff we do every day forms our ideas about ourselves and our culture. We see this in "Araby" on the individual level, that of the narrator. His thoughts are changed (change is the hallmark of dialectics) when he engages in social and economic exchange. He becomes cynical about his religious/romantic quest; he has cynicism for the whole notion of buying and selling at the bazaar. Although "Araby" is not frequently cited as a "Marxist story," the narrator is clearly alienated and changed by his experience in the marketplace. He is awakened, rudely, from his previous "false consciousness"—this previous consciousness when he did not feel alienated, or at odds with some dialectical oppositional force.
("False consciousness" is another Marxist term which means that one group dominates an oppressed group's consciousness—brainwashes them—basically so they stay in their place. The oppressed group or person is awakened when he/she realizes how the brainwashing is occurring.)