Both the condition of albinism (also known as an albino, although in some circles it is considered a derogatory term) and an animal having white hair are hereditary traits. This means that both conditions are the results of the combination of the genetic traits of both parents.
To put it simply a white haired animal is just that. An animal that's hair is white. It is no different than a zebra having black and white stripes or a raccoon having a black "mask" of fur around his eyes. For that matter (since we fit into the animal kingdom) it is the same as having blond hair or blue eyes. An animal that has albinism (or albino) has a complete lack of pigment in its skin and hair. This is caused by a lack of melanin in the cells. Animals with true albinism will have no coloring at all, including in the eyes. Most of these animals will have very pale, usually pinkish eyes.
Melanin is a pigment that helps to protect cells from the damage caused by the sun. In humans, precautions can be taken with sunblock and other measures. Animals in nature do not have this luxury.
How does albinism occur? We all remember studying basic genetics in high school. The experiments done by the pioneer in genetics Gregor Mendel with his pea plants taught us the way that certain genetic characteristics will behave. In any type of creature there are both dominant and recessive genes. They act in just the way the name implies. Most of the time the dominant genetic traits will overpower the recessive ones.
In rare occasions both sets of the recessive (non-dominant) genes with combine and create an offspring that only exhibits those traits. Albinism is a situation such as this. Because the dominant genes will almost always be prevalent in an offspring, the occurrence of an animal with true albinism is truly a rare occurrence.
The rarity of such animals has always been their downfall. In nature, they lack the natural camouflage needed to blend in. As an example think about an albino crocodile (one that has the condition of albinism). When it is floating in the water trying to look like a log (which is one way that they hunt) standing out as white as a vanilla ice cream cone will not get him very many meals.
Mankind has also noticed the rarity over the ages. Long before the science of genetics was even a glimmer in the eye of a shaman, albinism was still found in nature. Many times magical properties were connected with these animals. Because of this they would be hunted down and their body parts used as medicine and to show the "magic" of a holy man or Emperor.
You could, in truth, call these animals freaks of nature. Not because it is a bad thing, but because it is a rare thing. Having both sets of weaker genes combining to create an offspring is very, very rare.