Some dog breeds have short tails, and no, they're not born that way. They are born with a normal sized tail which is docked (cut, amputated, however you like to look at it) within days of birth. The amount that is left is dictated by the breed standards. It's quick, bloodless and done without anesthesia and a minimum of discomfort to the puppy.
Tail docking can also be done when the dog is older, but this requires anesthesia, surgery, antibiotics and pain medicine.
Some people who don't take their puppies to the vet do home jobs on their puppies. This involves placing a tight band at the section of the joint where they want the tail to end. It takes time for the tissue to die and the tail to fall off. The older the dog, the more pain is involved, and the smell is horrible. We've fixed our share of these home jobs gone wrong.
I was working a rare closing shift when a couple came in with an older puppy that had a home job done on its tail. Whoever did it had banded the tail in sections, and all but one section had fallen off. It was hanging precariously.
Further questioning of this couple determined that the man had found the puppy and its mother in the parking lot at work. They'd never had any kind of pet before, were financially strained, but wanted to help the puppy as best they could.
The boss sees the puppy, surgical amputation is nixed because of the cost, and the couple opt for antibiotics and washing the tail and applying antibiotic ointment. The boss gives an antibiotic shot and I'm directed to show them how to clean the tail and apply the ointment.
So the husband holds the puppy with the tail over the sink. I squirt some scrub on the tail, and begin to gently wash the tail.
The tail fell off. IT. FELL. THE. FUCK. OFF. In the sink.
The husband was horrified, the wife turns white and leaves the room. I yell for the boss, who as usual is done with them and on to the next thing. He tells me to continue with cleaning the tail. Fortunately, there is no blood and other than a little raw spot which will heal quickly, what's left looks like healthy tissue and the puppy will be fine. I gently finish and the couple leave with their new dog. I wish them well, their good deed is going to get expensive.