In electricity, a conductor is a material like copper or silver which readily conducts electricity, while an insulator is the opposite: A material which doesn't conduct electricity well. In other words, a conductor has a very low resistance, and an insulator has a very high resistance. A semiconductor is a material that lies somewhere in between: It will sort of conduct electricity, but has a fair amount of resistance. Although not commonly cited as a "semiconductor", carbon exhibits this property: Electricity can conduct through carbon, but carbon does have a significant resistance, and much of the electrical energy will be lost as heat energy when it passes through carbon. Not coincidentally, carbon happens to be used frequently to make resistors.
The electrical conductivity of an element is related to how many valence electrons an atom of that element has. An atom is orbited by electrons, but not all of these electrons orbit in the same place; there are different layers of space surrounding the nucleus of an atom in which electrons can exist. Each of these spaces is called a shell. Usually, electrical engineers are most interested in the outermost shell of atoms. The outermost electron shell is called the valence shell, and electrons which exist in the valence shell are called valence electrons.
Once again, it is no coincidence that the semiconductor elements all have the same number of valence electrons: Four. The common semiconductor elements are carbon (C), silicon (Si), germanium (Ge), and antimony (Sb). All of these are "Group IV" elements, meaning they have four valence electrons.
When people talk about "semiconductors" in the PC era, they are usually talking about silicon. Silicon is the most common element on planet Earth (about 28% of all matter on the planet is silicon), so it's relatively easy to come by, but for chip-making purposes, it's not as cheap as you might expect: The silicon used in chips must be super-refined.
In any case, in a basic configuration of pure silicon, the silicon atoms actually form a very tidy grid. The silicon atoms share covalent bonds, which are chemical bonds formed when two atoms share two electrons with each other (i.e. one electron from one atom enters the valence shell of the other atom, and one electron from the other atom enters the valence shell of the first atom). Each silicon atom, then, is bonded to four others: One above it, one below it, one to the left, and one to the right. Each atom has eight electrons around it (because each bond adds an additional electron to the atom's valence shell).
What makes semiconductors like silicon particularly useful is how readily their conductivity can be changed by doping. On its own, silicon is only a mediocre conductor, but the conductivity of it can be changed quite significantly by doping, which is the process of embedding a very small amount of atoms of other elements within the silicon. Typically, only about one in a million silicon atoms are replaced with atoms of some other element, but that one in a million makes a surprisingly big difference.
Even more interesting is how doped silicon responds to the application of electrical voltage. It turns out that by applying different electric charges to doped silicon, you can actually change how conductive or resistive that silicon is. It is this effect which made the invention of the transistor possible.
What affects this tendency is the existence of carriers within the silicon. A carrier, for the purposes of semiconductor basics, is an electric charge within the mass of silicon that's free to move around. There are two types of carriers: Electrons and holes. A free electron, or "carrier electron" is one which is not bound to any one atom, but simply drifts around inside the silicon mass, moving from one atom to another. Carrier electrons, like all other electrons, have a negative charge. Similarly, a carrier hole is the absence of an electron. Unlike electrons, holes are not actual physical matter, but their effect within semiconductors is similar to that of electrons, so holes are often spoken of as if they exist, even though a hole really is, quite literally, nothing. Since an electron has a negative charge, a hole, which is the absence of an electron, has a positive charge. Free holes, like free electrons, are not bound to any one place, and will drift around within the silicon.
To create free electrons, you add a donor element. A "donor" in this sense is an atom with more than four valence electrons (usually five). Group V elements like phosphorus (P) or arsenic (As) are typically used for this purpose. When an atom of silicon is removed from the silicon crystal and replaced with a Group V atom, the Group V atom forms covalent bonds with the four silicon atoms around it. This results in four of the donor's valence electrons being bonded... But what about the fifth valence electron? It doesn't get bonded to anything. Although it will usually stay attached to its parent Group V atom, this attachment is fairly loose, and the electron can easily be induced to wander around the mass of silicon. The electron is said to be "free". Adding several donors in this way results in a mass of silicon with a bunch of free electrons sitting in it. Because electrons have a negative charge, the silicon is called n-type silicon.
Conversely, to create p-type silicon, you replace a few silicon atoms with a Group III atom, one which has only three valence electrons. Common Group III elements include Boron (B), Aluminum (Al), and Gallium (Ga). This Group III element will be called an acceptor, which is the opposite of a donor, because while a donor adds electrons, an acceptor receives electrons. When the acceptor atom is embedded into the silicon grid, it will only be able to form covalent bonds with three of its neighbors. The final neighbor will have no bond, and there will be a "hole" between the acceptor and that neighbor, because there will be no electron there.
In both cases, the majority carrier is the carrier type which is more prevalent in a mass of silicon (either electrons or holes). If electrons are the majority carriers in a mass of silicon, the silicon is n-type silicon. If holes are the majority carriers, then you're looking at a piece of p-type silicon.
When introducing a semiconductor like silicon, you'll often hear talk of the semiconductor's intrinsic properties, meaning the properties that are inherent to the very nature of that semiconductor.
One intrinsic property that is important in semiconductors is the intrinsic carrier concentration. This is the number of carriers that exist within a given volume of the substance. This value is actually dependant on temperature; as the temperature of something changes, so does the intrinsic carrier concentration. The symbol ni (a lowercase letter n, followed by a subscript lowercase letter i) is used to represent the intrinsic carrier concentration.
A couple of other important symbols to know are those representing the equilibrium electron concentration, and the equilibrium hole concentration. Just as the intrinsic carrier concentration of a substance is affected by temperature, so too is the number of free electrons and holes. In the accompanying symbols, a subscript letter o is used to represent a state of thermal equilibrium (i.e. a state in which the temperature is not changing). The symbol no represents the equilibrium electron concentration, while po represents the equilibrium hole concentration.
Another interesting point is that in a pure semiconductor, all three of these symbols are equal. The equilibrium electron concentration is equal to the equilibrium hole concentration, and both of these values are also equal to the intrinsic carrier concentration. Neat, huh?
ni = no = po
(Two other symbols you'll see within this context are Nd, representing the donor concentration, and Na, representing the acceptor concentration.)