One characteristic is the use of simple characters, often animals. For example, in 'The Lion and the Mouse', the big and powerful lion and the tiny mouse are easy to understand. This simplicity helps to convey the moral clearly.
They usually have a straightforward plot. In most fables, there is a problem or situation that unfolds quickly, like in 'The Fox and the Grapes'. The fox can't reach the grapes, so it decides they are sour. The quick plot makes the moral, about sour - grapes attitude, easily grasped.