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!)

Scientists learn about the layers deep within the Earth's crust by studying how seismic waves travel through the Earth.If there is an earthquake somewhere, seismagraphs in many locations will record it. By looking at the time of arrival of the main set of waves, and how the frequencies of the waves are arranged within the set, scientists can learn about the density and other properties of the layers. The relative differences in arrival times of the S and P seismic waves at several recording stations tell scientists about the different speeds those waves were traveling at, which in turn gives information about the density of the material the layer is made of, and how thick the layer is in several directions.
If you line up a rod of metal and a wooden dowel, and hit one end of each with a hammer and press your ear to the other end. Does the sound wave reach the other end of each rod at the same time? If you can find rods that are the same size but of different metals (or maybe try plumbing and electrical conduit pipes, which you can probably find in steel, copper, and maybe galvanized aluminum) hit one end of each with a hammer. Do they ring at the same frequencies? Why or why not?

Scientists can tell by observing the seismic waves that are recorded all over the surface of the earth from distant earthquakes. The seismic waves are reflected (bounced off) layers of different density, and they are refracted (bent) when they enter layers of different density. Some of them don't go through liquid at all (the S waves). Scientists have been monitoring earthquakes and studying the phases of seismic waves that arrive at different stations for - hmm, well I don't exactly know - but certainly at least the last 75 years, with more and more sophisticated equipment. Seismologists look at the little wiggles that are made by pens on paper, connected to seismometers, every time a wave from an earthquake anywhere in the world passes under their station. It is by studying many of these seismic records, for many years, and pooling all our knowledge, that we have been able to come up with a working model of what the inside of the earth is made of, where the boundaries between layers of different density and composition lie, and why we have earthquakes where we do.

Something you can do to model how seismologists "listen" to earthquakes is have a friend tap on a big table while you put your ear on the table at the other end. We put sensitive instruments in the ground that act like ears so we can detect seismic waves from distant earthquakes.

Excellent question! Scientists use waves to study the different layers of the earth. Usually, they use seismic waves, which are waves generated by earthquakes or nuclear-test explosions. The seismic waves are bent, sped up, or slowed down, or even reflected when they pass through the earth's layers. Different types of materials (liquid vs. solid, rigid vs. softer) determine the speed of the waves. Bending of the waves occurs at the layer's boundaries. So, scientists study the path and speed of these waves through the earth to decipher boundaries and the materials that make up the layers.

Scientists know about the layers of the earth by sending sound waves through the earth. Because layers of different densities (see earlier question) will allow sound waves to travel through them differently, we can study the sound waves to understand the layers of the earth.

Well, we can't go there, you are very right! We are restricted to deep mines (less than a mile) and also deep drilling (about 10 miles at MAX!)...but seismic waves generated when an earthquake occurs in fact travel throughout the entire Earth !! Hence by studying the rate at which these waves travel through the earth we can infer the density thickness and overall characteristics (composition) of the earth as a function of depth. When we do that we find that 33 % of earth is iron metal, and the remaining portion is silicate materials(rocks)

Geologists have known for about 100 years that the Earth is composed of four layers; the Crust, the Mantle, the Outer Core, and the Inner Core .
Scientists still argue about the makeup of these layers and exactly how each layer interact with the other layers. We are not even sure how the layers were formed but we have some theories.

Because we can not go to the center of the earth we have to find our answers otherwise.

This is what a geologist by the name of Andrija Mohorovicic did. He discovered in 1909 that earthquake waves near the surface moved slower than earthquake waves that passed through the interior of the Earth. He also noticed that the P (primary, first and strongest) waves that passed through the interior of the Earth did not do so in a straight line. These waves were bent or deflected by something!
What the scientist knew was that waves of all kinds move faster and straighter through denser, more solid objects.
So Mohorovicic came to the conclusion that the outside layer or Crust was made of less dense material (Rock) and the next layer, the Mantle was much denser. This would explain why the earthquake waves moved slower through the crust.
So by looking at the seismic waves from earthquakes the scientist learned about the crust and the mantle but they also learned about the outer and inner core.
To do this you have to look at a different kind of waves, the S (secondary waves) waves that also get released by an earthquake. These S waves are slower.
Beno Gutenberg , a German geologist, believed that the Outer Core must be made of a liquid because the slower S waves could not pass through this layer and in fact "bounced off" and were deflected many degrees off course.
The fourth layer, the Inner Core, is composed of very, very hot metals (iron and nickel) with pressures so great that the metals do not flow as a liquid, but are forced to vibrate in place like a solid.
Earthquake waves that reach this layer move at the greatest speeds because waves move through solids faster than through gases and liquids.
This is how we know that there have to be different layers. Otherwise the behavior of the different seismic waves would not make sense.