What do you call stories with morals?
It could be a fable or a parable.
Fable= story that teaches a moral lesson/principle, but often times uses animals or inanimate objects to carry the message through.
Parable= story that teaches a lesson, but uses human characters.
Example of parable- 'The Boy Who Cried Wolf'
Example of fable- 'The Tortoise and the Hare'