Adam's apple: This came from the idea that the protuberance on a man's throat was caused by a piece of forbidden apple from the Garden of Eden's Tree of Knowledge lodged in Adam's throat.
An apple a day keeps the doctor away: Derived from the old English saying, "Ate an apfel avore gwain to bed, make the doctor beg his bread." The name of the author of this most popular apple saying has been lost to history.
An apple for the teacher: Likely harkens back to the "apple polisher." The custom of "apple polishing" hails from the little red schoolhouses of yore, when young children sought to become the teacher's favorite with a gift of a bright, shiny apple. So says the ditty: "An apple for the teacher will always do the trick, when you don't know your lesson in arithmetic."
Apple of my eye: In ancient Greece and Rome, people thought the pupil of the eye to be a globe-shaped object, like the apple. The Anglo-Saxon word "aeppel" meant both "eye" and "apple." In addition to providing vision, the pupil was also regarded as the "window" to the treasured secrets within each person. Thus, the "apple of my eye" meant someone very beloved.
As American as apple pie: Through the 19th and early 20th centuries, apple pie became the symbol of American prosperity, causing one American newspaper to proclaim in 1902, "No pie-eating people can be permanently vanquished." The full expression is "As American as motherhood and apple pie." The phrase "For mom and apple pie" was also the stock answer of many soldiers in World War II, when journalists asked why they were going to war.
One bad apple spoils the whole bunch: First coined by Chaucer as, "the rotten apple injures its neighbors."
The Big Apple: The smoky jazz clubs of New York City were the favorite hot spots of the likes of Charlie Parker and other jazz greats in the 1930s and '40s. New York City was said to have "lots of apples on the tree" — that is, lots of places to play jazz.
Upper crust: In early America, apple pie was a favorite dish. But to save on lard and flour, only a bottom crust was made. More affluent households could afford both an upper and a lower crust, so those families became known as "the upper crust."
Sources: The U.S. Apple Association and Wikipedia