In Chapter 6, why does Gatsby introduce Tom as "the polo player"?
Gatsby reveals his contempt for Tom Buchanan when he calls him "the polo player." The men had met "two weeks ago," although clearly they knew one another in the scuffles for the affections of Daisy in the past. However, when they were unknowingly "introduced" to one another, "Tom had accepted the introduction as a stranger."
As the small talk continues, Mrs. Sloan wishes Gatsby would accompany her to dinner in New York. Gatsby remarks that he hasn't "got a horse. I used to ride on int the army but I've never bought a horse."
This is infomation is relevant to his later snide remark, calling Tom a "polo player." When it is Gatsby's turn to "introduce" Tom to the party guests, he "took them ceremoniously from group to group: "Mrs. Buchanan...and Mr. Buchanan --." After a instant's hesitation he added, "the polo player."
Note the two digs Gatsby gets in here: not only does he introduce Daisy first (and pauses before he is able to spit out Tom's vile name) but by calling him a polo player, he is implying that he is rich and soft, that his experience with horses did not parallel his own masculine experience of horses in the Army. He is trying to make Tom look as ineffectual and impotent as possible to Daisy.
Tom is a lot of things: a very rich man, a graduate of the quite prestigious Yale University, a member of an old and extremely wealthy family, the owner of a beautiful and sizeable estate in East Egg, a husband, a father, and a onetime polo player. When Gatsby introduces Tom as "the polo player" to guests at his party, he implies that this is the most important or interesting aspect of Tom's identity.
Normally, one would introduce someone by listing what they do for a living or something really important about them, but Gatsby doesn't, because he wants to belittle Tom. It is like damning him with faint praise: Gatsby implies that Tom does not really have any other notable qualities by using such an inane one as part of his introduction. Gatsby manages to insult Tom without actually saying anything bad about him.