“Mrs. Buchanan . . . and Mr. Buchanan——” After an instant’s hesitation he added: “the polo player.”

from F. Scott Fitzgerald – The Great Gatsby (Chapter VI) on Rap Genius

Meaning

While introducing Daisy and Tom, Gatsby needs something else to call them by to make them appear more interesting. Tom objects to the false claim that he is a polo player but the other guests seem to remember this false tidbit about him.

All of the characters in the novel are constantly conscious of their public image; Gatsby recites his false history, Jordan Baker is the cool sportswoman and Nick, in his first passage, attempts to portray himself as a man without judgement. Gatsby’s introduction of Tom as “the polo player” thus robs Tom of some of his power; it is Gatsby, not Tom, who paints a picture of the Buchanans to his party guests.

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