What fairytale elements are present in the book and what is the significance?

Coraline has a number of fairytale elements in the book such as a sense of the normal being twisted, or happy endings. Plus, Gaiman actually mentions fairy tales directly in the introduction before the book starts. He quotes Chesterton by stating, “fairy tales are more than true: not because they tell us that dragons exist, but because they tell us that dragons can be beaten.”
When writing about the fairytale nature of Coraline, that quote can help you out since fairy tales from older times are more about how characters react to dark things than they are about the idea that fairy creatures exist.
In many fairy tales it starts with the normal, like a visit to a grandmother, and then twists it in some way, such as revealing that she’s actually a wolf. Coraline does this with the protagonist’s parents. Coraline travels to another world where things seem similar to her normal reality at first, like copies of her parents, but then twists it so that her parents have buttons for eyes, and actually want to hurt her instead of helping her.
The story also has a coming of age theme where younger characters have to learn to think for themselves, about other’s feelings, and other aspects of becoming an adult. Coraline definitely does this, so it’s worth using that.