My niece is dying to know why squirrels move their tails when chattering? I have no clue. 
Basically for the same reason that they are chattering - to alert other squirrels in the vicinity that there is a predator or threat about. They are giving out both auditory signals by chattering and visual ones by flicking the tails.
Many small mammals and birds will scold a predator that they can clearly see and which is too far away to be able to catch them. They recruit others to help, a behavior which is called “mobbing.” The predator, knowing it has no hope of catching a member of the mob that is alert to its presence, and that it can’t hope to take any other prey by surprise as long as it is being mobbed, will usually go away.
Thanks for the prompt replies, ltfire and Colibri; you’ve put a smile on a young girl’s face. I tried to thank you two last night but the board had other ideas.