The core of the earth is mostly iron with some nickel and perhaps a little bit of some other elements. We can figure this out in large part based on Earth's gravity. The amount of gravitational force that exert is dependent on its mass. We know the size of the Earth and the density of the crust, so we can tell that for Earth to have enough mass to generate its gravitational pull, the inside must be much denser than the outside. Iron has about the right density for that. It is also one of the most common metals in the solar system, which we can figure out based on the composition of meteorites. Still, iron is less common near the surface of the earth than we might expect given its overall abundance, so it makes sense for the rest of the iron to be in the core.
The outer core is molten, but the inner core is solid. We can tell because of the way seismic waves from earthquakes and nuclear waves go through it. There are two main types of seismic waves: s-waves (aka shear waves) and p-waves (compressional waves). An s-wave is a side to side or up and down wiggle that does not go through fluids. Sure enough, they don't go through the outer core. P-waves are like the springing motion of a slinky. They can go through solids and liquids, and their speed is related to the density of what they travel through.
When p-waves travel through the inner core, they go faster than they would if it were molten, so we know it must be solid. The speed also tells us that the waves are not going through pure iron, which is why we think there are other elements like nickel. The other reason we can tell that the outer core is molten metal is that Earth has a magnetic field. This is generated by electrical currents created as the metal flows. If it were not metallic or if it were solid and unable to move, there would be no magnetic field.
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