Chrysotile is a mineral of the serpentine group that crystallizes in thin, flexible fibers. As you can see on this specimen from northern California, the thicker the vein, the longer the fibers. (See a closeup.) It is one of several different minerals of this type, suitable for use as fireproof fabric and many other uses, that together are called asbestos. Chrysotile is the dominant form of asbestos by far, and in the home it is generally harmless although asbestos workers must beware of lung disease due to chronic overexposure to the fine airborne fibers of powdered asbestos. A specimen like this is completely benign.
Serpentine rock, or serpentinite, is the metamorphic product of magnesium-rich basalt that is subjected to high pressures and fairly low temperatures. The coastal region of northern California is largely made of serpentinite, the result of seafloor crust being subducted beneath the North American plate, then rapidly raised to the surface as the tectonic setting of the area changed. Particularly good-looking pieces are known as California jade, although true jade is tougher and lighter colored. See some of the variety of serpentinite in the Serpentinite Gallery.
Chrysotile is not to be confused with the mineral chrysolite, a name given to off-green varieties of olivine.