Earthquake waves are sound waves, and sound waves travel at different speeds depending on what they are traveling through. Different layers of the Earth have different densities, and this causes the paths of earthquake waves to bend as they cross these boundaries just as light waves bend when traveling through a prism. As a result, by noticing where earthquake waves bend and where they don't, and how long they take to reach you depending on where you are on the Earth relative to the earthquake, you can get an idea of what the layers of the Earth are beneath the surface.

Earthquakes produce two kinds of waves, S-waves and P-waves. S-waves only move through solids, while P-waves can move through solids and liquids. Therefore, by looking at the region where S-waves can reach, we can see how large the liquid magma core of the earth is.

Also, by looking at the speeds of the waves, we can learn about density of the material, as it affects wave speed. (Think like how underwater, sound moves more slowly.) Here please click to see is a good diagram of S- and P-waves and how they move around the liquid core.