VetInfo lists many dog breeds alleged to have hair rather than fur, including poodles, bichon frise, Irish water spaniels, Portuguese water dogs and lhasa apsos. The simple difference between hair and fur is that fur grows to a certain length and stops, while hair grows nearly continually.
Dogplay.com explains that hair and fur are composed of the same biological substance, keratin. Both come from follicles that go through four stages of life: anagen, or growth; catagen, a transition phase; telogen, a dormant phase; and exogen, when hair and fur are shed. Hair stays in anagen, the growth cycle, for the longest. Fur, however, goes through anagen quickly and spends most of its life cycle in the dormant phase, telogen. For this reason, fur seems to grow to a certain length and stop.
Hair and fur in the telogen phase tends to shed "feathers," the microscopic scales that make up the exterior layer of hair. These scales, blended with dead skin and dried pet saliva, make pet dander, the allergen that triggers allergic reactions in people. Dogs that have hair instead of fur have less dander because fewer hairs are in the telogen phase at any given time. Health.com suggests that curly-coat dogs such as poodles are even better where allergens are concerned because the curly hair tends to trap shed dander.