Electricity is made using a variety of processes:
Electro-magnetic: This is something that takes mechanical motion and converts it to electricity. The common way this is done is with a generator that is turned using a steam turbine from a coal or nuclear power plant. But it could also be a hand-cranked generator, a car alternator, or even one of those "shake to charge" flashlights.
Electro-Chemical: This is a basic battery-- both rechargable or normal batteries. The electrical energy is released from a chemical process.
Thermo-electric: When you get two dissimilar metals and you heat up one side and cool another an electrical current is generated. I can't possibly explain this one very well so here is the Wikipedia link.
Photovoltaic: A solar cell. Converts photons directly into electricity.
Electrostatic: Basically, using static electricity to do something. Normally this would be a van de graaff generator. It is hard, but not impossible, to make something useful with this as an energy source.
Piezoelectric: Some materials are piezoelectric. If you strike/hit/vibrate the material it will generate a small electrical current. This technique is mainly used as a sensor rather than a power source, but there are some places where using it as a power source is becoming practical.
That's all I can think of. I'm sure there are more, but these are the main ways that electricity is generated.