A collision is elastic if the total kinetic energy of the colliding objects is conserved as kinetic energy (overall energy is conserved in all collisions, whether elastic or not). No macroscopic collisions are truly elastic, although some come close. A frequent cause of highly inelastic collisions occurs when one or more of the objects suffers a plastic deformation.
A deformation of a material is elastic if it changes shape due to an applied load, but that when the load is removed, recovers its original shape.
The word elastic is often used colloquially to refer to an elastomeric material such as rubber or cloth/rubber combinations.