Steam power plants, whether they are coal or nuclear, use heat to make electricity. They operate like a giant tea kettle, turning water into steam which is then used to turn a generator to make electricity. The only difference between coal and nuclear power plants is that nuclear plants use uranium as the fuel to produce the heat instead of coal.
In a nuclear power plant reactor, water is heated by a process called nuclear fission.
The steam then spins the turbines, which are tied to the generators. The generators then produce electricity.