The fibrous structure on a plane of fracture or on a damaged spot, will often reveal if a material which may contain asbestos is involved. What's more, asbestos was mainly used after World War II and before 1993. The period in which the material was applied, can therefore be an indication of whether asbestos could be involved. Knowledge of the products in which asbestos is often applied may also help to determine if a material could contain asbestos.
A definite answer to the question if the material is truly asbestos-containing can only be given by examining it with an electron microscope. This is also the only way to determine the type of asbestos you're dealing with, and whether different types have been combined in the material.