How do scientists learn about Earth's core?

The earth conducts seismic waves- - when an earthquake occurs, stations farther and farther away from the quake see the S and P waves propagated through deeper and deeper layers. By measuring the arrival time of the waves, the velocity of the waves can be found as a function of depth. There is clear evidence for several layers in the earth which both refract the waves and below which the velocities are different. This is a bit like holding a book under your desk while a friend (gently) taps the other end of the desk. If you listen closely to the sound (i. e. with your ear on the top of the desk) the sound changes noticeably if a large book is pressed up against the bottom of the desktop. You might also try to figure out how to tell if a golf ball is wound, liquid filled or solid- - without looking at it. (You can tell if you hit one!)