SparkED: How do power companies generate electricity?

In a variety of ways, usually, depending on the region. The vast majority of it is generated by magnetic induction, but some is generated by the photoelectric effect. I'll try to be comprehensive in this answer.

First, generation of electricity by magnetic induction:

The concept is relatively simple, and in practice it's surprisingly simple, too. While the generator depicted above is a pretty poor generator, it demonstrates the concept okay. The math behind magnetic induction is, in differential form (easier from an electromagnetics standpoint),

or in integral form (easier from a systems analysis standpoint)

Where B is magnetic flux density, phi is total magnetic flux through the coils, the big E is electric field and the curly E is voltage.

Here's how they do it in practice.
By far the cheapest per kilowatt-hour is hydro-electric power.

Quite simply, it uses gravity to naturally let water fall, and the moving water turns a turbine to generate electricity by magnetic induction. It's cheap per kilowatt-hour because there's no throttle time issues and no fuel. It can provide both baseline (low-power usage) and peaking (high power usage) generation. As long as it rains enough in the region, the reservoir will fill up without any human energy input.

A lot of our electricity is generated in coal power plants.

This power plant takes coal and burns it, using the heat to boil water and create high-pressure steam, which high pressure steam gets forced through a turbine, which generates electricity by magnetic induction.

Coal is pretty disgusting, by the way, but it's cheap unless you charge for the pollution. Coal power plants usually provide baseline power, since it requires incrementally more coal per extra bit of power. Coal for peaking is more expensive.

There are other systems in a coal power plant to make the process more efficient and have cleaner output, but those are a discussion for another question.

For peaking power, we often use natural gas generators.

It's similar to how the coal power plant works, except that it gets to double-dip on power generation -- first, the combustion reaction turns a turbine, and second, the hot exhaust boils water to turn a steam turbine.

Gas is expensive compared to coal, but the first turbine throttles up really quickly and the second within a few minutes, and it can be throttled up and down very easily compared to coal. As such, it makes for a great peaking generator.

In some areas, baseline power is provided by nuclear power plants.
Nuclear power is remarkably similar to the others, as it's a thermal power generator.

The major conceptual difference is that the heat comes from Radioactive decay. Nuclear makes an excellent, clean baseline power source, and the major issues are failsafe systems (prevent meltdowns) and where to store the spent (still radioactive, but not enough to work in this configuration) fuel rods. However, it isn't very good at throttling up and down, so it sticks to baseline power generation for the most part.

I believe the biggest obstacle to deployment of more nuclear power is NIMBYism -- "Not In My Back Yard" opposition.

Next up, we have wind power.

In principle, it's even simpler than a hydroelectric plant. For newer systems, it's pretty cheap. The wind turns some great big blades, which turn a generator. The most expensive part of wind power is probably the land area that it has to take up per unit of power. Another issue with wind is that in most places it can be pretty variable, meaning you can't at-will throttle it up and down, and it may not be producing power all the time. If you have a very diverse set of locations for wind generators connected to a grid, it can work well, but if you rely on a single location the generation is a little intermittent.

If you want to know a lot more about wind power, ask Michael Barnard.

Keeping with magnetic induction, some companies have been using concentrated solar-thermal power. The concept is, again, a thermal power system, but instead of burning fuel it concentrates sunlight on to a target to heat it up. Think of the way you used to burn ants with a magnifying glass, only switch the magnifying glass for a parabolic mirror, and make it a lot bigger, and that's how solar-thermal works. It comes in different forms, but the concept is the same in all cases.

Modern systems have a salt target that melts and stores the heat, allowing the power generation process to continue for several hours after the sun goes down.

Last, I'll talk briefly about photovoltaic power, which uses the photoelectric effect instead of magnetic induction.

The basic concept is that light can knock charge carriers out of a bound state in a material, if individual photons comprising that light have enough energy to do so. In a photovoltaic panel, we use a semiconductor p–n junction and make the light get absorbed in the depletion region of that junction where there are no native free charge carriers. The light "generates" charge carriers (knocks them off of the atoms holding them) and they diffuse to the electrodes. It generates DC power.

I personally have a lot of interest in photovoltaics. (See: Jacob VanWagoner's post in X-Ray Visions for an interesting lecture on conversion efficiency, Jacob VanWagoner's answer to Is solar power becoming more efficient? and many other things I've answered related to solar panels.) While they suffer the same limitation of not having 100% uptime as wind power, the two major advantages I see are distributed power generation that takes up no real usable space, and that the panels tend to generate the most at the time of highest demand -- the afternoon, when everybody is running air conditioning.

Which ones are used most? Depends on where you live.