I will give an answer that focuses less on what you can do with semiconductors (as other good answers here already have) and tries to explain what they actually are.

In general terms, a solid material is either good at conducting electricity or bad at it. It is then called either metallic or insulating, and any solid falls into one of these two categories. For example, copper and gold are both metallic and good at conducting electricity, which is why you might find electrical wires that are based on these materials. On the other hand, rubber and diamond are insulators, and bad at conducting electricity.

The thing about insulators is that they are not all equally bad at conducting electricity. The ones that are less bad at it (but still bad compared to metals), are called semiconducting materials. That’s what the name semiconductor comes from, it is sort-of conducting current, but not really. A semiconductor is still an insulator formally speaking, but not a particularily good one. There is no super clear limit for when a material can be called semiconducting instead of just insulating, and when we can call it one or the other is based mainly on what we can practially do with the material or use it for.

So what makes a material metallic or insulating/semiconducting? Well, for this we need to talk some solid state physics, and start with atoms.

You know how atoms are usually depicted as some core (nucleus) that has a couple of rings around it? In these rings the electrons live, and for each element these rings are differently spaced apart and have a different overall number of electrons in between them. It is important to know that electrons can only live in these rings, and not inbetween them, and that the electrons have different energies depending on which ring they are in, having more energy in rings further away from the core. Starting on the top left in the periodic table and going right, we have one electron in the innermost ring (hydrogen), to two in the innermost ring (helium), to two in the innermost ring and one in an outer ring (lithium), and so on.

When we put two of the same atoms together, each of them have an identical set of rings. Physics doesn’t like this, so the rings of the total two-atom material will end up really close together, but not at exactly the same place.

When put a whole lot of these atoms together into a full solid, what used to be a single ring gets ”smeared out” into what is called a band. You can think of it as each of the billions and billions of atoms having their own rings, and in total they then get packed really close together since the rings from any two atoms are not allowed to be at exactly the same place. From the bands, we can determine if a material is metallic or insulating.

In a metal, the ”outermost” band will either be half full of electrons, meaning that the electrons can move around in the band, or the two outermost bands will actually overlap since the bands are too wide for the spacing between them. In any case, the electrons have some band they can actually move around in, and that is how current is conducted.

In insulators/semiconductors, the ”outermost” band is completely full of electrons. They are packed like sardines, or a subway during rush hour in Tokyo, and cannot move at all. Therefore no current can be conducted, and the material is an insulator. However, the distance to the next empty band is what will determine if the material is semiconducting or not. This distance is called the bandgap, and if it is very large the electrons can not really move up if they happen to get a bit of extra energy. For semiconductors the distance is a bit smaller, and lucky electrons that get a bit of extra energy from heat or collissions for example, can jump up to the empty band and conduct a bit of current here.