There are a few different ways to generate electricity. I'll go over them and what they mean and how they are used.
1. Electrostatics. If you've ever seen one of those metal-ball-on-a-pedestal-super-static setups, that's what an electrostatic generator is. You have to somehow generate friction to build up electrons on one part, usually an electric motor, but can be done by hand generation. It is extremely inefficient as a means of generating electrical power (partly because it's not really predictable, and there is no stable waveform), but it was the first method of ever experimenting with electricity.
2) Electrochemistry. Essentially it uses chemical reactions (acids or bases reacting with different metals) to move electrons. We use it all the time -- they're called batteries. This was the first method of getting steady electrical current, and the first widely available use of electricity -- they used "Voltaic Piles" (invented by Volta) to power "Electric Torches" or flashlights. It may seem cheap to you, at about $5 per pack of batteries, but if you powered your home like that, then it is really expensive per kilowatt-hour. It's great because it's portable.
3) Magnetic Induction. This is by far the cheapest way of electrical production. When a magnet and a conductor move relative one to another (let's say . . . turning a magnet inside a wire coil), it induces an electrical current in the conductor. It would be extremely tricky, I daresay impossible, to directly generate DC voltage using this method, so AC is the standard way it is generated, in a sinusoidal wave (put y = sin x into your graphing calculator in radian mode to see). There are loads of other reasons we use AC, but one of the main reasons is that it is the cheapest to produce. How power plants work is essentially this -- get something to turn a turbine of some kind, and the turbine turns a magnet inside big coils of wire. And there are loads of methods to turn those turbines.
. . . -- Coal plants burn coal to boil water, and the steam turns turbines. This is the cheapest method of producing electricity in terms of cost per material and maintenance cost, as well as approval to build with environmental restrictions. Other fossil fuel plants do the same thing, just burning other stuff instead, but coal is generally cheaper.
. . . -- Hydroelectric plants use moving water to turn the turbines. Cheap materials, but more expensive to maintain (look up cavitation on Wikipedia, it's the way water breaks concrete and other materials), and environmental considerations have to be made much more than with other power plants.
. . . -- Nuclear power plants use the radioactive decay of Uranium (puts out a lot of heat) to boil water and turn turbines. It is cheaper in material per energy cost than coal, but much more expensive to build a nuclear plant, and people are freaked out whenever they hear "nuclear" or "radiation" (BTW, all light IS radiation), and nuclear technicians and engineers are more expensive to hire than standard linemen and electrical engineers.
. . . -- There is at least one power source that I've seen this in -- using a parabolic reflector (curved mirror) to focus sunlight on a water tank, heating and boiling the water to turn a turbine. The one I saw was in California.
. . . -- Ever seen those hand-powered flashlights and other stuff? That's electricity by magnetic induction as well. They just store the energy by some other means.
4) Piezoelectric effect -- Certain crystals have properties that if you put enough pressure on them or bend them enough, you displace charge and can get a very momentary current from them. In addition, if you apply a voltage to them, they will bend. Often piezoelectric crystals are used in sound generation, they are very frequently used as oscillators or LC circuits (because they behave like an inductor and capacitor together, either in parallel or in series depending on how you use them -- look up RLC circuits on Wikipedia to see what they are used for). Generally the voltages are too low and generating power with them would be too inefficient to use them for power generating purposes.
5) Thermoelectric effect -- When you touch two different metals together (usually zinc and copper), they charge oppositely. If one side of them is hotter than the other, then you can actually get a steady electric current going through them. Usually it's too small to use for power generation, but it is used in digital thermometers and temperature monitoring. However, for nuclear powered satellites, physicists just stick a blob of plutonium on one end of a thermocouple (plutonium's radioactive decay produces heat) and that's how they power the instruments when it's too far away from the sun.
6) Photoelectric effect -- It has to do with the nature of light, being an electromagnetic wave (with particle-like properties). Essentially if you put enough energy in a light beam and