What causes things to be elastic?

Elasticity is the ability of a solid to return to its original shape and size after it has been deformed by a force. All solids have some elasticity. Familiar materials that have elasticity include the springs in cars and the rubber in tennis balls. Solids return to their original shape and size if the deforming force was not too great. If the stress (deforming force per unit of area) applied to a solid exceeds the solid's elastic limit, the solid will keep its new shape. The elastic limit depends markedly on the type of solid considered; for example, a steel bar or wire can be extended elastically only about 1 percent of its original length, while for strips of certain rubber-like materials, elastic extensions of up to 1,000 percent can be achieved.
The elastic properties of many solids in tension lie between these two extremes. The different macroscopic elastic properties of steel and rubber result from their very different microscopic structures. The elasticity of steel and other metals arises from short-range interatomic forces that, when the material is unstressed, maintain the atoms in regular patterns. Under stress the atomic bonding can be broken at quite small deformations. By contrast, at the microscopic level, rubber-like materials and other polymers consist of long-chain molecules that uncoil as the material is extended and recoil in elastic recovery. For further information on rubber elasticity, please refer the URLs below:

http://www.matter.org.uk/matscicdrom/manual/rb.html

http://www.poco.phy.cam.ac.uk/teaching/A_Donald/rubbers.htm