The diagram below represents what goes on inside of most power plants. The steam is shot out under high pressure at a turbine which turns like a pinwheel does when you blow on it. The turbine is attached to a magnet so that when the tubine spins, the magnet spins as well. The magnet is inside a generator which has a lot of coiled electric wires. When the magnet spins inside the coiled wires, the magnetic field moves in relation to the wire coils. This movement causes the electrons in the wires to move, thus producing an electric current.
Sometimes instead of steam, water from a dam is used to turn a turbine, or wind is used to blow a windmill in order to produce electricity in much the same way as in the diagram. Another common way that many utilities produce electricity is to use the energy from the sun rays. When the energy from the sun rays are absorbed by a photo cell, it is converted into electricity and then usually stored in a battery for later use.