To “curry favor” with someone means to do something that will ingratiate you to them. For instance, “Joe pretended to be interested in the project in an attempt to curry favor with his new boss.” This phrase has an interesting origin. The first part, “to curry”, is an equestrian (horse-related) term, which means “to clean the coat of”. This process was completed with a currycomb, a comb made of rows of metallic teeth or serrated ridges and used especially to curry horses.
OK, but how did it come to be associated with “favor”, and why would anyone “clean the coat of” favor? Turns out that this one traces its roots all the way back to AD 1310 and a long French poem written by Gervais du Bus called Roman de Fauvel ("Story of Favel"). The protagonist in the story is an allegorical horse that is the incarnation of the sins represented by the letters of his name: flatterie, avarice, vilainie, variété (fickleness), envie, and lâcheté (cowardice).
From that poem comes the Middle French expression estriller Fauvel, which was translated into Middle English as currayen Favel, meaning "to curry Favel." Because Favel was a chestnut-colored horse (in Middle French fauvel means "fallow colored"), the expression literally means "to rub or smooth down the chestnut horse." But Favel was not just a horse. He was a humanlike symbol of dishonesty in just about all of its forms. Therefore, currayen Favel figuratively means "to stroke or fawn on the evil chestnut horse to get some benefit in return.”
By the 15th century, the expression had become to curry favel. By the 16th century, folk etymology had replaced the still-unfamiliar word favel with the familiar and coincidentally relevant favor. Today, then, to seek to gain favor by flattery or fawning over someone is to curry favor.
