Earth’s interior is composed of solid and liquid layers of different compositions. Scientists know this because s-waves (elastic waves that shakes the ground back and forth, perpendicular to the direction the wave is moving) and p-waves (primary waves) generated by earthquakes behave differently as they pass through Earth’s interior.

S-waves cannot pass through liquids, and do not pass through Earth’s core because the outer core is liquid. P-waves can pass through both solids and liquids, but the speed that p-waves travel changes as they pass through layers of different rigidity (compositions and densities). Generally, p-wave velocity increases with depth and increases with increasing rigidity of a material. Therefore p-waves travel fastest through the solid, iron inner core of Earth.

Waves travel through different materials differently. A great example we can imagine is touching the surface of water and watching the ripples spread, and then touching the surface of jell-o and seeing how the `ripples" spread -- the wave that our hand makes in the water is very different than in jell-o because one is fully liquid and the other is a gel.

The same ideas apply to layers within the earth. The inner core of the earth is solid, so waves travel faster there than they do in the mantle, which is molten (semi-solid, semi-liquid). There are some waves that travel through solids but not through liquids, so they can tell us which parts of the earth are solid/liquid.

The different behavior of waves through different parts of the earth is actually how scientists first discovered that the inner core is solid!

Different layers inside of the Earth have different density, being made of different material, at different temperatures. Density and temperature determine the speed of sound, so the speed of the waves.