Asbestos is so dangerous because it can cause a variety of lung disease:
Asbestosis - Effectively scarring of the lungs that stiffens them and reduces the victims ability to breathe.
Lung Cancer: It's associated with asbestos, but may not be a direct result. The subjects who developed it were usually heavy smokers as well.
As the previous poster said, the other very dangerous complication is mesothelioma. Even tiny amounts of asbestos can cause it to occur, the dust left on clothing handled can cause it in people who haven't directly handled asbestos. It's very quick to develop, is usually not detected early and there aren't any truly effective treatments.