My son want a puppy like yesterday. We will be going to an animal donation center tomorrow. My wife is adamant that it has be be a dog that doesn't shed fur. I don't know if such a breed exists, but I know some shed less fur than others.
This answer is part of Pet's Spring Cleaning Campaign. This question is old, but this answer will still help people with the same problem.
You're looking for a hypoallergenic dog.
A hypoallergenic dog means this:
A hypoallergenic dog breed is a dog breed that is purportedly more compatible with allergic people than are other breeds.
In reality, there isn't such thing as a true hypoallergenic dog, but a dog that is far less allergenic than others.
There's no such thing as a hypoallergenic dog breed, although some individual dogs may cause fewer allergy symptoms than others. Many people think that pet allergies are caused by a dog's or cat's fur, but the real source of pet allergies is often a protein that's in the saliva and urine of dogs and cats.
Things to look for
You want to look for short coats and hair rather than fur.
I would also recommend reaching out to the shelter or breeder to ask if you could spend a few hours with the dog, to see if your allergies react.
The above two options are backed up by helix.northwestern.edu:
“These dogs have hair instead of fur. Their hair grows much as a human’s does,” Becker said. “We usually advise people with allergies to spend a few hours with our dogs just to see how they react.”
Other than that, the only thing you can do is to Google the breed (if you know what it is).
Examples of hypoallergenic dogs:
Most to all poodle crosses are somewhat hypoallergenic (including the Goldendoodle, Labradoodle, Aussiedoodle, and my personal favorite - the Bernedoodle).
The following is a short list taken from here and here (which have more detail and pictures). More near hypoallergenic dogs can be found here and here. There are many more wonderful hypoallergenic dogs out there, so don't limit your research to this list:
(links are to Wikipedia)
Poodle (Miniature, Toy, and Standard)