This Question is Beyond the Scope of This Site
This subject takes much, much more room to precisely answer than a single post can do here. People get degrees to answer this question. A single post here will not answer your question with exactness. A few years studying physics and then going on to study material science- that will answer your question as precisely as it can be answered. For now, I am going to describe this using very broad statements, but give you some sources to further your learning.
Quantum Mechanics, Band Gaps, and Materials
If you have ever used the Schrödinger Wave Equation to model physical systems, you should notice that particles can exist at different energy levels. If you model atoms, you will notice that atoms have certain rules of how to order electrons in particular energy levels. You find there are forbidden energies for electrons, and that only one exists at a particular energy level (although two may exist with opposite spins at approximately the same level). If you have multiple atoms in a close space, like those found in solid matter, these energy levels form bands of allowed energies and disallowed energies. In semi-conductors, these bands sometimes carry special names. These bands come in different flavors as well, which generally indicate how that crystal can be most easily used.
Anyways, semiconductors have bands which are "close" enough together that electrons can jump bands without needed large amounts of energy. The energy could come from, say, a regular electric current from a wall or which is used in modern electronics. These semi-conductors are then doped with various other types of atoms. The dopant you use depends on what you want the semiconductor to do. The introduction of impurities in the crystal allow for an excess or dearth of electrons, giving "wiggle room" for electrons to jump up/down the bands of the material as they travel through the material.
It turns out that, if you're tricky with how you dope things, you can make a p-n junction. These junctions are very useful, because it allows current to sometimes flow. This make them tiny transistors, which can be used to develop the procedures used for computing. This is the basic idea behind the smallest component of modern computers. (Although there used to be computers which stored data in arrays of pieces of iron which would be magnetized, but those have been replaced.)
Generally, the electrons we shove around with current inside these junctions have a very short way to go, and electrons can travel pretty quickly. Quick enough, anyways, that we can switch these junctions to the tune of billions of times per second. The ability to control doping operations has greatly improved the density (by decreasing the size) of p-n junctions, therefore processing power and speed have also increased. This is actually a problem we're encountering in cutting-edge chip development; these junctions are too small! We can make these junctions so small that we have trouble keeping electrons in certain states due to random thermal fluctuations and tunneling! (Not that current computing systems don't suffer the effects of thermal fluctuations and tunneling- these effects are just small enough that they effectively don't matter!)
A Note About Silicon
Silicon is not always the "medium of choice" for semiconductor material. As an example, if you want a high-temperature chip, you may choose Yttrium or some other crystal to dope. If you want something that actually performs better, GaAs is a popular crystal. The point is; there are other types of semiconductors, and silicon is not special because it is the only thing we can use.
Silicon is often chosen just because it is so readily available and fits most applications good enough. Our planet has silicon present in a lot of places. It is also relatively easy to harvest and purify to make components. Dirt itself is sometimes more expensive than silicon!