At the beginning of The Catcher in the Rye, Holden had just returned from a disastrous trip to New York City. Holden was the manager of the Pencey fencing team, and he accompanied them for their meet with the McBurney School. However, the team returned without having competed: All of the team's equipment was lost when Holden left it on the subway, and his angry teammates "ostracized" him all the way back to Pencey. While he was in New York, Holden did manage to purchase a "red hunting hat," with "very long peaks" for just one dollar. Holden decided to wear the hat backward, which looked "very corny"; but, nevertheless, he liked his appearance better that way.
I have to admit that bullgatortail has answered this question very well, but I am happy to provide a few more details for you. The simple answer to your question is that Holden bought a hunting hat that was both red and had "long peaks" during the famous Fencing Team's trip to New York City.
I suppose we could equate this purchase to Holden's desire for things (and people) who aren't "phony." This hat certainly isn't phony and is quite utilitarian (at least, for a hunter it would be). The hat is described as follows:
[Holden bought] a red hunting hat with very long peaks.
Holden is especially proud that the hat only cost him one dollar. Further, and again in an attempt not to appear "phony," Holden appears "corny" while wearing the hat backwards in order to flatter his appearance even more. Meanwhile, Holden tries to redeem himself a bit after losing all of the equipment on the subway. (The poor Pencey Fencing Team!) I suppose we can consider that Holden now has only his good looks, his wit, his attempts at being legitimate (and not "phony"), ... and his red hunting hat.