How does a generator make electricity?

It's funny you should ask, because we know how they work and we will be glad to tell you!!!
To answer that question we need to define what a generator is. In electricity generation, an electric generator is a device that converts mechanical energy to electrical energy.
In 1831–1832, Michael Faraday discovered the operating principle of electromagnetic generators. The principle, called Faraday's Law is that an electromotive force is generated in an electrical conductor that circles a varying magnetic field. He also built the first electromagnetic generator, called the Faraday disk, a type of homopolar generator, using a copper disc rotating between the poles of a horseshoe-shaped magnet. It produced small DC (direct current) voltage. A homopolar generator is basically a DC electrical generator made up of a electrically conductive disc rotating perpendicular to a static magnetic field. A generator is just an electric motor working in reverse. This all may sound confusing right now, but will become more clear later!

I think it will be easier to understand a generator if we understand a motor first. Basically, an electrical motor is an extremely tight coil of copper wire wrapped around an iron core that is free to rotate inside a powerful permanent magnet. When an electric current is introduced, the coil basically becomes an electromagnet with a current going in the opposite direction of the permanent magnet. The two currents interact, and the tight coil of copper wires spins. This spinning can power anything from a small toothbrush to an electric train.

So how are motors and generators alike? A generator is the opposite of a motor. So instead of introducing the electric current to spin the coils as described earlier, we spin the coils to introduce the electric current. I think this concept will be much easier to understand as explained in this example form "Suppose you have an electric toothbrush with a rechargeable battery inside. Instead of letting the battery power the motor that pushes the brush, what if you did the opposite? What if you turned the brush back and forth repeatedly? What you'd be doing would be manually turning the electric motor's axle around. That would make the copper coil inside the motor turn around repeatedly inside its permanent magnet. If you move an electric wire inside a magnetic field, you make electricity flow through the wire—in effect, you generate electricity. So keep turning the toothbrush long enough and, in theory, you would generate enough electricity to recharge its battery. That, in effect, is how a generator works".
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electric toothbrush with motor described

The concept of electrical generation using electromagnets is that instead of permanent magnets, two electromagnets opposite to each other induce the magnetic field around the rotor, which turns and produces electricity.

There are many types of generators that produce electrical energy, all of which use magnets or magnetic fields to some extent.
A more simple generator to understand is the Dynamo. The Dynamo was the first generator to use electromagnetic principles to produce power for industries. A Dynamo Generator consists of a stationary magnet that provides a magnetic field for the entire generator, and rotating windings that rotate in that magnetic field. With smaller machines, a permanent magnet would provide the magnetic field. With larger machines, one or more electromagnets, also referred to as field coils, would provide the magnetic field.
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One of the first Dynamo generators A modern day Dynamo generator

The dynamo generator is not a practical generator in todays society because it requires a great amount of physical labor for very little electricity in return. They can commonly be seen on the back tires of bicycles that produce electricity to illuminate a light on the back. More practical generators are ones that utalize steam power to turn the generator's Axel.
steam turbine
steam turbine

Modern steam powered generator

Dynamo Demonstration

General information