In biology, all mammals have hair, no other kind of animal (e.g. bird, insect or fish) does. There are many types of hair, some soft and some hard, some coarse and some fine, etc. Some plants have hair too.
A dense coat of fine, soft hair is called fur: bears, cats and mice have fur. Although fur is made of individual hairs, by convention we don't call it hair.
A dense coat of very fine curly hair is called wool; most sheep, some rabbits, and some extinct species of mammoth and rhinoceros have (or had) wool. Again, by convention we don't call it hair.
Animals whose hair isn't fur or wool are said to have hair: humans, pigs, dogs, even some breeds of sheep. Generally this is because the hair is too sparse or long (human), too long and straight (hair sheep) or too coarse (horse, dog) to be considered fur or wool.