Much of this story is told in stream-of-consciousness style, following first the flow of the maid Lily's thoughts and then the flow of Gabriel's thoughts. When Gabriel arrives at the Christmas party and offends Lily, he feels awkward, self-conscious, and in need of validation. He views the world around him in terms of his own ego. He worries about making his Christmas speech, thinking:

He would fail with them just as he had failed with the girl in the pantry. He had taken up a wrong tone. His whole speech was a mistake from first to last, an utter failure.

Later, after his speech, he feels better, and his eyes are bright with happiness. He feels loving towards his wife, Gretta.

He sees himself in a mirror in the hotel room when he and Gretta are alone together. This is significant because initially, he seems at peace with what he sees, a figure in a "broad, well-filled shirt-front." Significantly, too, he is "puzzled" as he "always" is by his expression. This suggests he lacks self-knowledge.

After he hears Gretta tell of her first love, Michael Furey, he mentally rearranges the self he just saw in the mirror. He becomes filled with self-loathing, mentally describing himself in scathing terms as "ludicrous," "nervous," a "sentimentalist," a "fatuous" object of pity and full of "clownish lusts." He moves from not understanding himself to interpreting himself as someone who other people find laughable and contemptible.

Through Gabriel's changing perceptions of his reflection in the mirror, Joyce shows how much Gabriel depends on the opinions of others, especially his wife, to validate his self-worth. Like a good modernist, Joyce also shows the subjectivity of external reality, such as the reflection in a mirror, which is dependent on our moods, perceptions, and frailties.

Here are the passages that describe the moment that Gabriel sees himself in the mirror (they don't have page numbers because they're taken from eNotes' excellent online version of the text): 

As he passed in the way of the cheval-glass [Gabriel] caught sight of himself in full length, his broad, well-filled shirt-front, the face whose expression always puzzled him when he saw it in a mirror, and his glimmering gilt-rimmed eyeglasses. 

Later, Gabriel muses on this experience in a fairly negative way: 

[Gabriel] saw himself as a ludicrous figure, acting as a pennyboy for his aunts, a nervous, well-meaning sentimentalist, orating to vulgarians and idealising his own clownish lusts, the pitiable fatuous fellow he had caught a glimpse of in the mirror. 

All of this happens around the time Gretta, Gabriel's wife, tells him about Michael Furey, the young man she loved in her youth in Galway. The story is a romantic one, and Gabriel quickly realizes that he is not his wife's true love after all and that he has never captured her affection the same way this young man did. Within this context, Gabriel's bitter evaluation of his reflection becomes very important. Gabriel's reflection in the mirror becomes a physical manifestation of his realization that he is not as important as he seems. Indeed, the reflection underlines Gabriel's sudden recognition of his own ridiculousness and relative insignificance. In that case, Gabriel's sight of himself in the mirror becomes a pivotal turning point in the text, as it is the beginning of the epiphany that ultimately destroys Gabriel's illusions about himself and forces him to reassess some of the most significant aspects of his life.