While some dogs short tails are natural, called bobtails, many dogs you come across will actually have docked tails.
        
The term 'docked tails' refers to a dog's tail that has been intetionally cut off as a puppy, often times to fit breed standards. Examples of both can be seen below.
 
    
 
         
Image Sources
  
Australian Stumpy Tailed Cattle Dog
  
Brittany (short) Brittany (long)