The inside of an apple is called the "exocarp," according to New England Apples, which is the part that is eaten. This part of the apple is inside the skin, but outside of the core, also known in common terms as the flesh of the apple.
Apples start as flowers, and once they are cross-pollinated by bees that carry pollen from one blossom to another, the flowers become fertilized and grow into apples. First the petals fall off, then seeds are developed and the ovary grows, which becomes the core of the apple. The outer layer of the ovary (core) becomes the exocarp.