Where did we get our evidence about the Earth's structure?
From p waves and s waves
P waves and S waves are types of siesmic waves generated by the movement of techtonic plates.They can also be artificialy generated by exploding bombs.These waves have different properties.For example. p waves can pass through all mediums: solids,liquids and semi solids while s waves cannot pass through liquids.Thus they create certain 'shadow zones' where no siesmic waves are found.
They also are different in their method of propogation and shape.
Thus by knowing the rate of slowing down of the waves in materials tested on earth, the composition can be known by doing a direct comparision.
Also by knowing the angles of p and s wave shadow zones and then doing a little complicated geometry, the radius of each of the layer and thus the thickness can be found