Rubber elasticity refers to a property of crosslinked rubber:  it can be stretched by up to a factor of 10 from its original length and, when released, returns very nearly to its original length.  This can be repeated many times with no apparent degradation to the rubber.  Rubber is a member of a larger class of materials called elastomers and it is difficult to overestimate their economic and technological importance.  Elastomers have played a key role in the development of new technologies in the 20th century and make a substantial contribution to the global economy.  Rubber elasticity is produced by several complex molecular processes and its explanation requires a knowledge of advanced mathematics, chemistry and statistical physics, particularly the concept of entropy.  Entropy may be thought of as a measure of the thermal energy that is stored in a molecule. 
Common rubbers, such as polybutadiene and polyisoprene (also called natural rubber), are produced by a process called polymerization.  Very long molecules (polymers) are built up sequentially by adding short molecular backbone units through chemical reactions.  A rubber polymer follows a random, zigzag path in three dimensions, intermingling with many other rubber molecules. An elastomer is created by the addition of a few percent of a cross linking molecule such as sulfur.  When heated, the crosslinking molecule causes a reaction that chemically joins (bonds) two of the rubber molecules together at some point (a crosslink).  Because the rubber molecules are so long, each one participates in many crosslinks with many other rubber molecules forming a continuous molecular network. 
As a rubber band is stretched, some of the network chains are forced to become straight and this causes a decrease in their entropy.  It is this decrease in entropy that gives rise to the elastic force in the network chains.