Loading a structural element or a speciment will increase the compressive stress until the reach of compressive strength. According the properties of the material, failure will occur as yield for materials with ductile behaviour (most metals, some soils and plastics) or as rupture for brittle behaviour (geomaterials, cast iron, glass, etc).
For long structural elements (such as columns, truss bars) with aspect ratio greater than 2:1 (length:section), increase of compressive stress leads to buckling (at lower stress than the comressive strength), namely the elastic instability.
Compressive stress has stress units (Force per Area), usually with negative values to indicate the compaction. However in geotechnical engineering, compressive stress is represented with positive values.