Dr. Inge Lehmann discovered the makeup of the Earth's inner core by studying how an earthquake's waves bounced off the core. It was previously thought that the core was made of liquid, surrounded by solid mantle and a crust. Lehmann found that the Earth's center is made of a solid inner core surrounded by a liquid outer core.
We cannot physically get to the Earth's core because it is too hot, the pressure is too extreme and the distance is too far to explore. Scientists must study seismic waves generated by earthquakes to understand the inner workings of our planet. Inge Lehmann was a Danish seismologist who was studying the waves of a 1929 earthquake when she found them acting inconsistently with how they should have behaved if the core consisted of liquid surrounded by solid mantle and crust. She wrote a paper in 1936 theorizing that these waves had traveled to the core and then bounced off some kind of boundary. This theory was confirmed in 1970 when more advanced seismographs detected waves deflecting off this solid core.