I actually spent several years studying the composition of the core for my MS research. I wrote the following a few years ago for a similar question; the numbers may be out of date by now, but our fundamental understanding of the core has not radically changed.

We haven't drilled down anywhere near the core; the deepest well on Earth is a bit over 12 km deep, whereas the core-mantle boundary is 2890 km down. Here's what we know for certain: we have an excellent handle on the density profile of the core, and we know that the outer core is liquid and the inner core is solid. This has been determined from seismic studies. S-waves do not penetrate the outer core, indicating that it's liquid. The density jump at the inner/outer core boundary (along with a few other factors) indicates that the inner core is solid.

From studying meteorites, we also know that there was a huge component of iron-nickel alloy in the primeval Earth; the only possible repository for this metal is the core. There are several competing theories as to how the iron got to the center of the planet, but they all involve massive partial melting of the early Earth. This process, regardless of the specifics, is called differentiation, as it's the time when the core differentiated itself from the mantle.

The biggest gap in our understanding of the core is its temperature range. We have some constraints from the melting curve of iron and from phase transitions in the mantle believed to occur just above the core-mantle boundary. However, the average temperature at the core-mantle boundary is still considered to be somewhere around 3500-4500 °C - a range wide enough to hinder many research projects.

We also know (from a combination of seismic and high pressure/temperature laboratory studies) that the core is slightly less dense than just pure iron. (The nickel component doesn't have a significant affect on the core's density). This is due to an additional light element contribution. The most likely elements, due to geochemical and cosmochemical factors, are O, Si and S (H and P are possible, but less likely). My research centered around the possibility of Si in the core, and I found that there could be about 6-9% Si down there depending on the temperature at the core-mantle boundary.