Compression testing is a way of determining how something will react when it is squashed, or when a force is applied to it in compression.  It is performed to:

Assess the strength of components e.g. automotive and aeronautical control switches, compression springs, bellows, keypads, package seals, PET containers, PVC / ABS pipes, solenoids etc 
Characterise the compressive properties of materials e.g. foam, metal, PET and other plastics and rubber
Assess the performance of products e.g. the expression force of a syringe or the load-displacement characteristics of a tennis ball