Electricity is generated by a change in a magnetic field near an electrical conductor. Normally, this is achieved by winding larges amounts of wire into a coil, and spinning this inside a magnetic field. This field is often generated by another fixed wire coil, or in the case of some smaller generators (such as bike dynamos), fixed magnets.
This basic principal is used in power stations fueled by a variety of different fuels. In most power stations where the fuel used gives off heat, boilers are used to produce steam, and this then drives turbines coupled to generators. This is true of coal, oil, and nuclear. Natural gas powered plants are the exception, as they used gas turbines to directly spin the generators. With hydroelectric and wind power, the generator is mechanically linked to turbines or propellers that are spun by the elements.
The odd man out is solar power. This uses the special semiconducting properties of deliberately impure silicon of induce electrical currents to flow when the silicon is hit by protons.
Countries differ in their choice for power generation. The USA still relies on coal for nearly half of its electricity. Mountainous countries like Switzerland and Norway often have significant hydro-electric generation. Around half of the power France generates is nuclear. Each method has different drawbacks, and the relative costs of different methods vary depending on location. The ever rising global oil and gas prices have returned coal to significance in the UK, and prompted an increase of interest in Nuclear power. Solar power has generally remained a small and specialist due to the high cost of the cells, although production of cells has increase significantly in the last five years.
For the long-term, it seems likely that nuclear and renewable energy will between them dominate the market. Nuclear power has remained a sensitive option ever since Chernobyl, but has huge potential for cheep energy. The high capital cost per megawatt is has kept renewable energy as a minority power source for many countries, but as electric prices increase, its market share is growing. Coal is likely to be a medium term fix for many countries, but the cost of controlling the pollution will probably gradually reduce its attractiveness.