According to an article in The Atlantic, apple seeds should not be eaten, although a person would have to consume a very large number of apple seeds for it to be fatal. Apple seeds contain amygdalin, which releases hydrogen cyanide when digested.
The amount of cyanide released from apple seeds is small, which means that accidental swallowing of apple seeds is not likely to be fatal, even if a person eats quite a few apples. Cyanide is best avoided at any dose, however. The apple core is safe to eat, excepting the seeds, and consists only of a thin fibrous band. The amount of the apple that the average person throws away is much larger than the actual core, resulting in a significant amount of wasted food.