Seismology (from the Greek seismos = earthquake and logos = word) is the scientific study of the movement of seismic waves through the Earth, as a result of natural or man-made activity. The field includes studies of earthquakes and similar variants as seaquakes, activities of volcanoes, landslides and movement caused by tectonic plates.
Earthquakes (and other earth movements) produce different types of seismic waves. These waves travel through the earth (soil and rock), and provide an effective way to "image" or "see" events and structures in the Earth's subsurface.
One of the earliest important discoveries was that the outer core of the Earth is molten. Pressure waves pass through the core. If the core was not in a liquefied state, ransverse or shear waves that shake side-to-side would be recorded as they respond to the presence of rigid material.
The process of mapping subsurface features is a specialty field called seismography or seismic interpretation. Seismic waves are generated in controlled experiments using dynamite and other explosives which generated waves within the earth. These waves are convolved to yield seismic data which have been used (successfully) to map geological features in hydrocarbon-bearing rocks, faults (cracks in deep rock), rock types, and a lot of other features in the earth.
Using seismic tomography with earthquake waves, the interior of the Earth has been completely mapped to a resolution of several hundred kilometers. This process has enabled scientists to identify convection cells, mantle plumes and other large features of the inner Earth.