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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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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