Tensile properties are the most important single indication of strength in a material. The force necessary to pull the specimen apart is determined, along with how much the material stretches before breaking.
The elastic modulus ("modulus of elasticity" or "tensile modulus") is the ratio of stress to strain below the proportional limit of the material. It is the most useful tensile data because parts should be designed to accommodate stresses to a degree well below this.
For some applications where almost rubbery elasticity is desirable, a high ultimate elongation may be an asset. For rigid parts, on the other hand, there is little benefit in the fact that they can be stretched extremely long.
There is great benefit in moderate elongation, however, since this quality permits absorbing rapid impact and shock. Thus the total area under a stress-strain curve is indicative of overall toughness. A material of very high tensile strength and little elongation would tend to be brittle in service. ASTM D-638.