Seismic waves that travel through the core move at a velocity that depends on the properties of the material the waves move through. The best fit is with iron, Fe. A second reason is that Fe is by FAR the most abundant metal in the UNIVERSE. Many meteorites have significant amounts of iron in them in the Native state.

It may be made of a mixture of other magnetic metals however there are several reasons to believe that the core is made of predominantly iron. First, iron is one of the most abundant elements on Earth. Secondly, asteroids are composed predominantly of iron. Scientists believe a major portion of early Earth formed by coalescing asteroids so by extension one may infer the interior is also composed of iron. It is so hot in the Earth's interior that we know it is composed of some liquid metallic material. Now different elements exhibit different densities and if allowed to remain in a liquid the elements will separate depending on their density the heavy ones to the bottom and light ones to the top. Have you ever seen salad dressing separate in the bottle; with the oil below and water above? The Earth's interior has been hot so long there would be reason to believe this process of density separation would have reached equilibrium and hence the thought that Earth's outer core is predominantly composed on one element.

Some of the most convincing evidence for an iron core composition comes from what we know about gravity and energy waves. We know Earth's size and gravitational pull therefore we can infer Earth's density. From Earth's density we can estimate what elements must compose the Earth; an iron core just happens to estimate Earth's mass the best. Now from energy waves, geologists use seismometers to measure movements in Earth's interior (e.g. Earthquakes), These energy-waves form compressional and shear waves. From seismometers around the globe we can see the shear waves disappear when they reach the depth's of Earth's outer core. This tells us that the outer core is composed of liquid material. This is because liquids cannot be sheared, ever try pushing water? It just moves out of the way. Now the compressional waves on the other hand actually slow down. That is because liquid is much less compressible than the material above partially molten rock. Now from understanding how energy waves travel through materials we can estimate at what speed will compressional waves move through different materials. It just happens that the speed at which the compressional waves move through the outer core is really close to the speed we would estimate for liquid iron. When the energy waves reach the inner core we can see the shear waves appear and the compression waves accelerate sharply. Similarly from understanding the speed at which these waves travel through materials we can see these speeds are really close to those estimated for a solid iron composition.

Now this does not entirely rule out different elements in the core. There may be other trace (small) amounts of other elements. However from what we see in asteroid composition, gravity of Earth, and how energy (seismic) waves travel through the core we have reason to infer the core is made predominantly of iron.

We know that it's iron chiefly because we know how dense it is. The density is measurable by how well it transmits sound waves, as well as by measuring the gravitational field of the Earth: if the Earth were made of solid rock, for example, gravity would be half as strong at the surface as it is. If the core were made out of lead, then gravity would be stronger.

Also, iron is the most common reasonably dense metal in the universe (because of how nuclear fusion works in stars), so it makes sense that the core is mostly iron.