How does nuclear energy work?

Chain reaction fission, energy produced, heats water, spins turbine, electricity.

This is basically how a typical fission reactor works.

In the reactor core there are the fuel rods, control rods, and moderator.

The fuel rods are usually an isotope of Uranium or Plutonium. A chain reaction is started by a neutron colliding with a nucleus of the fuel. This causes the nucleus to become unstable, and it will decay, causing three more neutrons to be released, along with some energy. These neutrons will cause further decay from other nuclei.

The energy produced heats up water outside of the reactor core, boiling it, so the steam rotates a turbine to produce energy by electromagnetic induction.

To control the rate of the reaction, there are the control rods and moderator.

The control rods are typically made from boron, and can be raised or lowered from the reactor core. They absorb neutrons so that fewer chain reactions occur. The lower into the core the control rods are placed, the slower the reaction will occur. Fully lowering the control rods will bring the reaction to a halt.

The moderator is often water, and it is used to get the neutrons to reach their "thermal energy". The neutrons collide with the moderator, this slows them down. After around 50 collisions, they will be at an appropriate energy to collide with a nucleus and cause fission, rather than being deflected.