Tension is when a material is stretched, while compression is when it is squashed. Different materials have varying abilities to withstand these forces, with elastomers exhibiting high elasticity, metals providing strength and ductility, and composites offering enhanced properties. Understanding tension and compression is crucial for designing structural elements like columns and beams, and mechanical testing methods like tensile and compressive testing help evaluate material behavior characterized by parameters like yield strength, elasticity, and stiffness.