Elasticity is the ability of the hair to stretch and then to spring back to its original state. Elasticity is important when doing chemical treatments, styling, or just maintaining healthy hair.
If the hair does not have elasticity, it can stretch to a point where it can break. To test for elasticity yourself, take a strand of hair and gently stretch it. If it springs back, it's fine, if it breaks, however, do not attempt any chemical applications.