A fairytale is a magical story for children, like "Cinderella" or "Rumpelstiltskin." If you read a story that begins, "Once upon a time," chances are it's a fairytale.
A story doesn't have to include any fairies at all to be considered a fairytale. Fairytales are typically full of magic spells, princesses, talking animals, evil queens, witches, and goblins. Disney would probably not be in business if it weren't for fairytales. You can also use the word to describe some part of real life that seems almost magical, like a "fairytale romance."