The border between the outer core and the inner core is about 5,100 km below the surface of the Earth. (The crust is usually between 3 and 70 km thick, being thicker in the continents and thinner in the oceans, so you can figure out the distance between the bottom of the crust and the inner core by subtraction.)

The inner core and the outer core are made up of similar stuff chemically (both are made mostly of iron, with a little nickel and some other chemical elements)--the difference between them is that the outer core is liquid and the inner core is solid. As you get deeper inside the Earth, both the temperature and the pressure (due to the weight of all the overlying rocks) get higher. The outer core is liquid because the temperature is so high that the iron melts. Although the inner core is even hotter than the outer core, it's solid because the pressure is so high that it squeezes the atoms together so much that it turns into a solid.

You might be wondering how it is that we can tell that the outer core is liquid and the inner core is solid. This is because a very large earthquake releases waves of energy called seismic waves that can travel all the way through the Earth and be detected by earthquake-detecting machines called seismometers all the way on the other side of the planet. It turns out that an earthquake will release several different kinds of seismic wave, some of which can travel through both liquids and solids and some of which can only travel through solids, and by studying very carefully where on Earth you can detect these different kinds of waves from a single earthquake you can figure out the different layers of the Earth and what they are made of.

I hope this answers your great question! I am glad that you are curious to learn about the fascinating planet we live on!