Significant exposure to asbestos fibers causes asbestos-related lung diseases. "Significant" usually means you were exposed for at least several months to visible dust from the fibers.
Asbestos fibers are very small. When you breathe in, they can get stuck deep in your lungs. The fibers remain in your lung tissue for a long time and may cause scarring and inflammation. This can lead to pleural plaque and widespread pleural thickening, pleural effusion, asbestosis, lung cancer, or mesothelioma.
Generally, asbestos-related lung diseases develop 10 to 40 or more years after a person has been exposed to asbestos.
Being around products that contain asbestos isn't a danger, as long as the asbestos is enclosed. This prevents the fibers from getting into the air.