In the Americas, Australia and Europe (and a few other places: West Africa, the Congo Basin, Tibet and southern India), sheep have the skinny little tails we're familiar with. However, in most of the Old World--the rest of Africa, all the Near East and Central Asia, northern India, Mongolia, most of China, even Madagascar--sheep tails look like beaver or kangaroo tails.
These big tails store fat, and you might think them a natural adaptation to a harsh environment, like the camel's hump. Not so; all wild sheep have thin tails. The fat tail is a characteristic that ancient shepherds bred into sheep. Nomads didn't have any oil, and if they wanted to fry something, it had to be in sheep fat.
But why did they breed for a fat tail? Because they wanted soft fat, which is easy to melt.
An animal can metabolize its fat only in liquid form. As a result, fat stored near the skin has to have a lower melting point than fat deposited deep in the body, because it's exposed to the (usually) lower temperatures of the air. The tail was the ideal place to encourage sheep to deposit their fat, because it's almost entirely surrounded by the air and can develop a convenient slab of soft fat an inch thick.
Incidentally, tail fat isn't as disgusting as it may sound. Since it has a low melting point, it doesn't leave the tallow-like coating in your mouth that regular lamb fat does. It's sort of like thick bacon fat . . . with a muttony aroma, of course.