From Wikipedia:
Earth's inner core is Earth's innermost part and is a primarily solid ball with a radius of about 1,220 km (760 mi). (This is about 70% of the Moon's radius.) It is believed to consist primarily of an iron–nickel alloy and to be approximately the same temperature as the surface of the Sun: approximately 5700 K (5430 °C).
How do we know what the size of the inner core is?
Bonus Points: How did we come up with iron-nickel as being the believed constituent for the core?
We know the the size of the inner core through seismology. From my answer to this question: How are subsurface wave speeds determined without subsurface sensors?, we can determine the speeds of the different layers of earth. Pictured below is a diagram of raypaths going through the earth from the 1994 Northridge Earthquake in Southern California:
(image from http://serc.carleton.edu/NAGTWorkshops/geophysics/seismic11/overview.html)
As you can see the earthquake causes many raypaths, some of which go all the layers of Earth. From Huygen's Principle we know that there are infinitely many ray paths, meaning that there is a raypath, depending on location, that
and arrives at the same seismometer (probe that measures vibrations, or seismic waves in this case). Depending on the composition of these layers, the ray paths will have different arrival times. The difference between these arrival times are important, we call them lag times, which seismologists can use as a proxy for distance. The lag time between the 3rd and 4th raypath I mentioned above could be used as a proxy for the radius of of the inner core, but we probably would not get a good answer from that. More over, we use this seismic data along with other data types to constrain its size.
We can use gravity data to understand get the mass of the earth. See this question for how that can be achieved: How is the mass of the Earth determined?
Using the mass of the earth, its size, and assuming that density increases with depth, we can form a seismic wave model (in the first question I linked) which would give us a more accurate lag time to distance conversion.
We also know that Earth is made up of the same stuff as the Sun, by examining its composition through the light spectrum.
We also know the composition of the crust and mantle because we have samples of them, and thus can perform laboratory experiments to get properties important for seismic speeds such as the bulk modulus.
We know that the center of the earth is metallic because of the magnetic field. It was the Trela model that first proposed this. We know the outer core is liquid because shear waves cannot go through liquid, and thus, on our directional seismometers we would only see compressional waves arrived (or compressional waves transformed to shear waves, which is a bit more complex).
Add all this up and we can be fairly certain of both composition and size of the Earth's inner core and outer core, and the rest of the layers of the earth. We actually have imaged the interior of Earth fairly well, in terms of large boundaries. Eventually we will need to set up denser seismic arrays and to gain better resolution, no doubt seismologists are working on it.
Scientists used the seismic waves created by earthquakes bouncing off the core to map out the approximate size of the earth's inner core.
The materials that constituted the core were guessed with the thinking that because it was once liquid, the heavier elements like iron and nickel were able to sink down into the center. It probably even has vast amounts of the heaviest elements, like gold, platinum and uranium.
Reference: Earth’s Inner Core
In 1906, Richard D. Oldham found that the increasing speed of seismic waves with depth within the Earth holds only down to 2890 km below the surface. Deeper than that, the mechanical waves (sound) propagate much slower, suggesting a different rock nature. Because this distinct material did not transmit shear seismic waves, it became clear that this core is liquid. This is what we call the Earth's core. Its size is known because the farther your seismometer is from a earthquake hypocenter, the deeper the registered seismic waves have travelled: since the velocity of sound increases with depth down to the core, seismic rays are refracted and they curve back towards the surface. Because the velocity-depth relation breaks abruptly at the core-mantle boundary, this is clearly detected in seismographs. I link this example of how to use seismic refractions to determine the size of the core.
In 1936, Inge Lehmann, using the same technique, found that the center of the core is indeed nearly-solid, because she detected weak shear waves travelling through it [2] using highly-sensitive seismometers in New Zealand. This has become known as the inner core.
References:
[2]: Lehmann, I., P'. Publ. Int. Geod. Geophys. Union, Assoc. Seismol., Ser. A, Trav. Sci., 1936, 14, 87-115.
The previous answers are correct in that geophysics and crust-mantle samples are the main tools for sorting out the composition of the core.
In addition, don't forget that meteorites are the solid 'left overs' of planetary formation, and we have many thousands of meteorite analyses. Most are 'rocky', but some are palasites (mixed rock-iron), or 'irons', whose composition is basically iron with a smaller percentage of alloyed nickel and a few other heavy metals. Even allowing for the fact that average meteorite composition does not necessarily yield 'average Earth' composition, the presence of so much nickel-iron in meteorites is a strong pointer to the likely core composition of the Earth, and is entirely consistent with clues from geophysics.
