As with any other smaller storm, a hurricane's storm clouds rotate counterclockwise due to forces in the atmosphere like gravity and pressure. These forces compete with calm air's natural tendency to rotate clockwise.
It's a complicated system, though. The rules I've just stated only apply in the earth's Northern Hemisphere. In the Southern Hemisphere, everything is reversed, with hurricanes (and all other storms) rotating clockwise! That's because below the equator, calm air's natural tendency is to rotate counterclockwise.
Confused? It helps to know that calm air (air that is not rising and does not contain any storms) has a tendency to naturally flow in a direction caused by the earth's rotation on its axis. This flow is more gentle and gradual than most. When a storm forms, its winds are stronger and more violent, essentially defeating this more tranquil flow and turning the air in a different direction.
By the way, the force that causes the initial turning of calm air across the earth's surface is called the Coriolis Force. Here again, things get complicated, as the Coriolis Force is actually not a force at all, but an apparent turning of winds, based on the curved surface of the earth and the land rotating beneath it.