The outer layers are pretty easy - scientists can go look at and pick up rocks from different parts of the crust to study their composition, structure, and order as depth below the very surface increases. The inner layers are more difficult to study and much of our knowledge relies on indirect measurements based on how sound waves, most commonly those caused by earthquakes, travel through the earth. Essentially, by measuring how waves take to travel through a certain distance of Earth's insides, scientists can work out details of the material through which the waves are traveling. This is discussed extensively in the ScienceLine responses to this question , which was asked multiple times in multiple forms already. We can also infer that there must be a liquid (at least partially anyway) core made of metal because Earth has a magnetic field.

Even though we can only take samples from the outer layer of the Earth, valuable information can be gleaned. The prevalence of certain elements (e.g. iron) in the rest of the solar system relative to that in the crust, in combination with the average density of the earth vs. that of the crust, leads scientists to conclude that the inner core must be iron.

BBC also has a very good article on this topic.

This is a great question!

There are lots of ways that geologists know what is inside of the earth and how it is structured. The most basic structure can be determined using seismic waves. When an earthquake happens, the earthquake produces three main types of waves:
P waves, S waves, and surface waves.

Scientists figured this out during the middle of the 20th century, as nuclear bombs can produce the same types of waves. Geophysicists (scientists who study the physics of the earth) set up detectors all around the globe, and whenever an earthquake happens, it gets recorded on these detectors. Because S and P waves have different behavior (S waves are slower, and P waves can't travel through liquids) , geophysicists can map out the inside of the earth as the waves travel through the different parts. I've included a really great diagram that shows this below, and have also found a short video explaining how waves move through the earth:
seismic wave paths.

Since P-waves don't travel through liquids, the P-wave shadow zone indicates that there must be a liquid part of the interior of the earth. By mapping where the waves end up, we can see that the outer core is liquid, and we can track how the waves refract (bend) through the different layers (e.g., crust, upper mantle, lower mantle, etc.) to determine their densities and what they are likely made out of. Even with all of that, though, there is still a lot that we don't know about the inside of the earth , like its composition (we know generally what the chemistry of the earth's interior is like, but scientists argue a LOT about specifics). I hope this has answered your question, and that you have more great questions about Earth Science!

Sound waves travel at different speeds depending on the density of the material they are traveling through. If waves are traveling through something that is changing the speed at which waves move, then the waves will change direction as they cross this boundary. Earthquakes make noise, and geologists build machines that can detect these sound waves as they travel through the earth. By comparing the results from many different listening devices at different locations, we can reconstruct what layers the earth consists of.

We use physical simulations to try to guess at what the layers with their densities are. So, for example, we know that the outer core has about the same density as a mixture of molten iron and nickel, and the inner core has a density of solid iron-nickel alloy. These, combined with observations about how common what elements are in the solar system and in the universe, lead us to surmise that these elements are probably what the core is made of.

To understand different rock layers near where we live on the surface of the Earth we have drilled to about 10 km deep. Doing this helps us understand shallower rocks like those we would find oil or natural gas in. Sometimes mountain building can expose layers that we usually don't see. That's why there are some rocks in California at the surface that used to be in the Earth's mantle, the layer bellow the crust.

To look at deeper layers we usually have to use different types of signals coming from layers in the Earth. For example, when earthquakes happen deeper in the Earth they vibrate the ground. We call the vibrations seismic waves. The waves move faster through some rocks than others, but we can then find the layers because layers have different speeds for the seismic waves. There's also another way to look at layers through the way they carry electricity, which we can use to tell the difference between layers based on how well they carry electricity.

When scientists can't directly measure something, like digging up rocks from beneath the earth's surface to directly see what is there, they use indirect measurements. Geologists and Geophysicists are the scientists who study rocks, and properties of the earth. The tools they use for studying the layers of the earth are called "seismic waves". They have tools to emit seismic waves (like sound waves) through the earth, and measuring how fast or slow they go. How fast the wave goes depends on the material the wave moves through. Think of when you yell into air, compared to when you yell under water -- the sound waves from your mouth travel differently and sound very different. Sound waves also travel differently through different types of rock. Geologists and geophysicists measure how the seismic waves travel through different rock layers to identify what the layers are.