While the mechanism by which it does it is not known, asbestos will increase your chances of getting lung cancer if you breath it in. Lung cancer is a disease of uncontrolled growth of lung tissue. Asbestos can also produce asbestosis, a scaring of the lungs similar to that caused by coal dust in "black lung" or dirty cotton dust in bysinosis. Asbestos also produces pleural plaques, an increased density of some lung tissue, as seen in x-rays.
Symptoms in general are changes in sense of well being, in body function or appearance, or in health that would suggest a recent exposure. There are no symptoms of asbestos exposure. Years after a substantial exposure has begun, a disease condition may become apparent. That is not a symptom of exposure, but it is a result of exposure.