I'm allergic to dogs and cats. I've had a lot of people tell me that I can have a dog if I find a species that has "hair" instead of "fur". I can't tell you how many times I've heard that. It's a myth.
"Hair" and "fur" are the same thing. A defining characteristic of mammals is the presence of hair on the body. The word "fur" is simply a term used to define hair on non-human mammals.
Cat's clean themselves by licking their fur. So they are coating their fur in their saliva. Also, some people can be allergic to the lick of a dog. Same thing applies.
I did read that there is a chemical(a mild sedative) that can be added to a cat's water (like 1 drop per gallon of water) which changes the chemical makeup of their saliva. The cat doesn't notice a difference or feel different, but their saliva is no longer an allergen. My girl and I are talking about getting a cat some day, and if we do we're going to look into this first to see if its a possibility.
Read more here about pet allergies: http://www.allergybegone.com/aboutpetdander.html