Despite being relatively close to us the inside of the Earth, and particularly the core of the Earth, is difficult to investigate. Primarily because we can't just look at it - and the deepest mines or boreholes are only 10km deep which is tiny compared to the 6,000km that is the Earth's radius. So everything needs to be logically deduced from the readings that we can take. Discussing what we know about the Earth's core and how we know it on In Our Time were Stephen Blundell (University of Oxford), Arwen Deuss (Utrecht University) and Simon Redfern (University of Cambridge).
Prior to the 19th Century the assumption was that the Earth was the same all the way through - rocks where we can see, so rocks everywhere. But in the 19th Century scientists realised that the theory of gravity required a denser Earth than is possible if it's just rocks and so they postulated an iron core. This was also the time when scientists began to be interested in how the Earth was formed. The consensus at the time was that it formed by condensation out of a hot cloud, and it was still cooling. This explained (at a time when radioactivity wasn't known) why it got warmer the further you went underground. So at the time the best explanation for the structure of the Earth was that it had a hot liquid iron rich core surrounded by a rocky shell.
However even in the 19th Century it was clear that there were problems with this explanation. If you spin an uncooked egg, it wobbles - so why doesn't the earth? During the 20th Century it began to be postulated that the core was two phase - a solid core with a liquid coating. One of the experts on the programme, Arwen Deuss, used seismological readings to show that this was the case. When there is an earthquake seismographs on the other side of the Earth detect the shockwaves that have travelled through the planet. Before Deuss's work it was thought that there was a shadowzone where no waves were detected because they had failed to pass directly through the centre of the Earth - so it was thought that the core was a different phase to the rest of the planet and the waves couldn't travel through it. Deuss showed that there are very faint delayed waves detectable in that shadow zone, and that mathematically the best model to describe how these waves are delayed and how they are diminished is one where the core is solid but it is surrounded by liquid. The seismic waves cannot travel through liquid in the same state as they travel through solid, and each transition between states uses up some of the energy in the wave. A wave that travels directly through the core will transition from solid to liquid to solid to liquid and lastly to solid again. As well as each transition using up energy it takes time (hence the delay) and changes direction (so the waves aren't in quite the same places you'd expect if they had no transitions).
The current theory is that the inner core is an iron crystal that is forming out of a less pure molten iron fluid around it. This iron crystal is about the size of the Moon, a fact which I find mind-boggling. The crystal is still growing and this is not a consistent process, sometimes it grows more quickly and sometimes more slowly. The experts said there is evidence of some sort of discontinuity that formed 500 million years ago, but no-one knows what caused it. The crystal is also split into two pieces. One of the experts made an analogy with the land/sea divide up here on the crust, but I didn't really follow that. The crystal is also different in the north/south direction as compared to the east/west direction - seismic waves take longer to travel east & west than they do north & south. It's not known why this is: perhaps to do with crystal alignment, or perhaps it tells us something about the shape of the core.
This solid iron crystal is rotating within the liquid it sits in, I think at a slightly different (quicker?) speed than the whole of the Earth rotates. It's this rotation that is the cause of our magnetic field (which is another piece of evidence in favour of the two phase theory). And the magnetic field is what protects us from cosmic radiation so in some sense you can say that the two-phase spinning core of the Earth is why there is life on Earth. The current theory is that Mars and Venus have cores that are too solid or too small to generate enough of a magnetic field to protect against radiation. That's an untested hypothesis, and so Deuss would like to put seismographs on one (or both) of the other planets to see what she can detect about their internal structure.
Bragg closed up the programme by attempting to encourage them to talk about practical uses that have come out of this blue-skies research - but it seems at the moment this is still in the blue-skies phase.