Why do dogs have tails?

Dog TailsA dog's tail is an extension of the spine. It helps them to keep balance when they run. Moreover, A dog's tail position and motion is incorporated as a component of a complex system of body language that domestic dogs use to show excitement or agitation. 

In Addition

In mammals a tail appears physically as an extension of the backbone ("Chordata"- zoological classification of animals that have a backbone) is the central upright balancing hanger of adjoined parts of a creature. 

 

If you reduce a tail to a Chordate, the most abstract is an Eel or Snake. 

A tail can be thought of as an extra limb thats' mechanics may or may not be particularly dexterous or manipulative by its usage upon objects(except in Possums and Monkeys though they prefer to do much of and any dexterous manipulations of objects using any of the other limbs they have and sometimes but rarely the limbs adjoined to their pelvis' in preference to the limbs adjoining the neck and shoulders). 

However, as a primitive simple or poorly functioning limb it only tends to commit protection of the sensitive genital area of creatures in land based animals when used as a limb for a function by the actual direct usage for manipulation of its being. 

Definitely in dogs' and cats', emotional expression is abstractly obvious(definitely related to) related their tails manipulation by the tails' manipulator muscles having positioning or posture movements effected upon it deliberately. 

Definitely also it is obvious that the backbone is a jarring(adverb) fastening against the shock waves of movement caused by usage of its adjoined limbs during traversal, of the tail will be sending back more shock waves amplified by the tails' position setting muscles' associate nerves and mechanical effect of gravity relating each shock cycle upon the fulcrum of the tail base. 

The resulting usage of the resultant traversal shock cycle can indicate anything from fatigue to a forced repositioning of the tail by an external or environmental influence, inclusive, how well balance is being maintained in some situations of movement. 

Tails appear to have their best usage in flight(running at top speed) whether as an environmental sensor(wind) or as a defensive sensor when fleeing (enemy touches its tail during running). 

More Input and Opinions:

1. Contrary to popular opinion, dogs have tails just because that is how God created them. God made dogs with tails, so they have tails only because of that reason.
2. Dogs have tails for a special reason: To operates mechanically and psychologically (and also behaviorally and emotionally) 
3. In nature, ecosystematics help it survive.While its tail can easily be seen in motion it is not really for emotional expression or attraction of other animals except in nurture of their own young because dogs have a narrow field of vision though wild dogs can see prey in either night light or day light. Neither is it in any direct exacting for balance as is thought of the mechanics of (e.g. kangaroos' tails) as i will explain. While dogs do demonstrate usages of their tails alike communication or balance, there are other animals that demonstrate it is not required to have a tail at levels of predator ability and sizes that dogs are.
4. A dog is a carnivorous predator that is thought to have been evolved from the Grey Wolf over the past 10 million years and as the classification predator implies it hunts game for its food. 
5. However, carnivorous predating game for food requires a set of inbuilt drives psychologically that enable the dog to survive called predatory instinct(a primary directive in Canis), of this has a special part of it called"chase instinct".
6. As you have probably seen both kittens and puppies and even bigger varieties of either Canidae or Felidaechase their own tails and snap their teeth at them and once in a rare time injure their own tail mildly.This"chase instinct" has an extremely valuable point that it is part of the predatory psychological drive, that being itcauses them impulsively to react at sudden movements with a rush of emotion or feeling to attack before it comprehends exactly the cause by recognition of the cause.This is a major part of their ability as a predator to survive.
7. Now, "why its tail relates to chase instinct".
8. Generally predators(quadruped mammal) all point their tail directly out behind them straightened when they start to chase.
9. When predators form groups or are alone to hunt but particularly in predators that catch prey by chasing these predators must move at immense speeds almost always to catch their prey.
10. This could be required in a forest at anything from 45 M.p.h / 70 K.m.h. to 55 M.ph. / 90 K.m.h., so a chase lasting twenty seconds could have the dog rush past deadly solid objects such as boulders or tree trunks up to 100 times(during the chase) or more in a fraction of a second at top speed. 
11. This naturally is extremely dangerous but the final most important point is that because they are moving at top speed they "cannot simply stop or adjust course instantly or quickly if wind strikes side on" "unless they are disturbed enough".Because they are effectively a sail at side on to the wind a strong gust will throw them into a tree if they do not realize to adjust to the sudden mechanical environment change.
12. Something must hit the safety switch and cause their starter motor instinct(chase instinct) to shift its attention to the problem using enough emotion overpowering the initial chase.When the tail is struck by the wind at a particular level(e.g. pushing the tail over sideways on its fulcrum at its base while at top speed) it tells the chase instinct to switch its attention to the danger.This is made easy to cause by wind due to the natural mechanics of the tail being easy to disrupt its position and the fur struck by the wind triggers touch that also assists the chase instinct in being distracted to the problem that it could be thrown off course.
13. When it has finally identified it as wind it will even have adjusted instinctively toward(into the wind) the side the wind is coming from by bending around at its tail(metaphorically). It will not do that completely as a grown experienced high speed animal but will have moved somewhat to that position and adjusted its propulsion directions into the wind a little according its natural reactive dexterity.
14. A contrast of prey chasing technique relating the tail can be found in Bobcats(Lynx Rufus) that have a naturally omitted tail(no tail, or extremely short tail) by their biological mechanics, whereby they take more prey by ambush or stalking(no chase, do not pursue) of requires pouncing(ambush) or prey being inside a circumference of 3 to 6 meters and the prey having less ability to accelerate(ambush, not an actual chase) if it detects the Bobcat before it attempts the attack.
15. A tail would be disastrous on a predator attempting ambush techniques as its primary method because of the reactive nature chase instinct has and the greater warning any movement would be to prey animals from psychologically triggered disruptions upon the predator that are beyond the predators control e.g. wind striking its tail, young following in proximity. 
16. However alike a rabbit on the point of having no tail(a Bobcat or Lynx), they have ears that mechanically will be sensitive to strong wind gusts.