How do we know that the Earth has layers, and how do we know what elements make up each of the layers?
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primarily the ricochet of earthquakes. they don’t go straight through but at certain angles they travel faster with less lose of amplitude(strength)
IIRC, the different layers have different densities that refract the waves from earthquakes in different ways. Studying this phenomenon allows you to infer the composition from the densities required to refract the waves.
Earthquake observatories are a thing that study the Earth just like astronomical observatories study the sky. Here is an example that uses a steel ball to create artificial earthquakes. https://youtu.be/vxfJbW6KDp4
As other have said, we know the boundaries and densities of the layers from the refraction, absorption, and reflection of seismic waves. [Here](https://www.youtube.com/watch?v=u6-mJ6TCi6o) is a seismic wave model showing an example of how waves propagate through the earth. You can see how different layers have different effects on the waves. The [first](https://en.wikipedia.org/wiki/Andrija_Mohorovi%C4%8Di%C4%87) [people](https://en.wikipedia.org/wiki/Inge_Lehmann) working on the math behind these models were solving differential equations without computers, even sometimes relying on making models out of strings and wire.
As for what minerals make up each layer, there are many decades of research on different topics that have informed our understanding of the deep earth. Magma from the deep mantle tells us the elemental composition. Some deep minerals are brought to the surface in the form of [xenoliths](https://en.wikipedia.org/wiki/Xenolith), so in a sense we have actual samples. We also have [diamond anvil cells](https://en.wikipedia.org/wiki/Diamond_anvil_cell) that can recreate the temperatures and pressures from deep in the earth and see what minerals could exist. Even more basic, we know the primary ingredients of the planet from meteorites, and we know the mass and density from gravity.
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