Compressional waves are also known as a longitudinal waves because of the way in which they travel through a medium. Compressions and rarefactions occur in the direction of travel, which is often visualized as the snapping of a slinky (see figure below).
In seismology, compressional waves are often referred to as Primary waves (or P waves). These are the first waves to arrive after an earthquake. Some of the ways in which P waves can be used include earthquake early warning systems, to better understand the velocity of various mediums underground, and to create seismic in order to image underground structures and geologic beds.
Unlike transverse waves, compressional waves can travel both through the ground and through the atmosphere. This is because both solids and fluids (the atmosphere and bodies of water) can be compressed. 
A great video by Dr. Russell at Penn State that shows how compressional waves travel through a slinky:
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