What does it mean when a cat's tail wags back and forth?
First, when the owner is around the cat, the cat will wag its tail because it feels comfortable; this is a sign of encouragement.
Secondly, when the owner pets the cat, but the cat doesn't want to be petted, wagging its tail usually means it's impatient.
Third, sometimes when the owner calls the cat's name, the cat will respond by wagging its tail. 
Fourth, if the owner has food that the cat likes, the cat will wag its tail to indicate that it wants to eat the food.
Fifth, when a cat is in a particularly stressful environment, it will wag its tail to relieve its tension. When a cat confronts another species, its tail will be very straight and its fur will stand on end, indicating that it is in a state of alertness and ready to attack at any time. At this time, you should pay close attention to the cat's condition.
When a cat holds its tail high, it means it's in a good mood. If it droops to the ground, it might be scared. Cats can be aloof, but if a cat holds its tail straight up with a slight curve at the tip when it first meets a stranger, it means the cat accepts the new friend.
It's still standing upright, but it's slowly swaying from side to side and keeping its eyes on you. This means the cat is happy and wants you to give it a small reward or feed it.
