here is how a BASIC nuclear reactor works (SIMPLIFIED)
a nuclear reactor consists of 7 main parts

they are:

1- fuel

2- moderator

3- control rods

4- coolant

5- pressure valves and pressure tubes

6- steam generator

7- containment

fuel: uranium is the most common fuel, usually small pellets of uranium oxide is arranged in tubes to form fuel rods that are arranged in fuel assemblies in the reactor core, there can be over 51000 fuel rods with each rod having approximately 300 pellets, each rod is 4 meters long and made of zirconium alloy, each fuel rod can burn in 5 minutes.

moderator: material found in the reactor core that slows down neutron release, it is usually water but can also be heavy water or graphite.

control rods: these rods are made from neutron absorbing materials such as cadmium, hafnium or boron, they can be inserted or withdrawn from the reactor core to control the rate of reaction or to completely stop the reaction, secondary control rods are in the coolants, usually liquid boron and its concentration in water can be adjusted overtime as the fuel burn up.

coolant: a fluid consisting of water or deuterium, or a gas consisting of helium and carbon dioxide, or a liquified coolant consisting sodium, sodium-potassium alloys circulating trough the core to transfer heat from it, a nuclear reactor has two or four primary loops driver by 6.6MW electric pumps or steam, each electric pumps weights about 110 tons.

pressure valves and pressure tubes: usually a robust steel vessel containing the coolant or the moderator, but it also may be a series of tubes containing the fuel and conveying the cooling through the surrounding moderator.

steam generator: part of the cooling system where the high-pressure primary coolant bringing heat from the reactor is used to make the steam from the turbine in a secondary circuit, its basically a heat exchanger used to convert the coolant into steam using the heat produced in the reactors core.

containment: the structure around the reactor to protect it from outside elements, it is usually a 3 meter thick steel or concrete wall.

there are 6 main types of reactors.

they are:

PWR (pressurized water reactor)

BWR (boiling water reactor)

PHWR (pressurized heavy water reactor)

AGR (advanced gas cooled reactor)

LWGR (light water graphite moderated reactor)

FNR (fast neutron reactor)

PWR: this is the most common type out of all other nuclear reactors, PWRs use water for both cooling and moderating, it has a primary circuit where water flows through the core at very high pressures and a secondary circuit where steam is generated to drive the turbines.

BWR: BWRs are very similar to PWRs, except there is a single circuit where water is at lower pressure so it can boil in the core, the reactor is designed for 12-15% of water to be at the top part of the reactor as steam, the steam passes through the drier plates and to the turbines.

PHWR: PHWRs are one of the oldest designs, they use natural uranium oxide as fuel so they need a more efficient moderator in this case heavy water, the PHWR produces more energy per kilogram of mined uranium than all other designs but produces much more used fuel per unit, the moderator is in a large tank called the calandria penetrated by several hundred horizontal pressure tubes which form channels for the fuel cooled by heavy water under high pressure.

AGR: these are the second generation of British gas cooled reactors using graphite moderator and carbon dioxide as primary coolant, the fuel is uranium oxide pellets enriched to 3% in stainless steel tubes, the carbon dioxide circulates through the core and past steam generator tubes outside it but still inside the concrete-steel vessel hence the integral design, control rods penetrate the moderator and a secondary shutdown system involves injecting nitrogen to the coolant.

LWGR: it employs long 7 meters tubes running through the graphite moderator which is cooled by water that is allowed to boil in the core like in a BWR, fuel is low enriched uranium oxide pellets made up into fuel assemblies 3.5 meters long, with moderation largely due to fixed graphite excess boiling simply reduces cooling and neutron absorbtion without inhibiting the fission reaction.

FNR: some reactors don't have a moderation and instead utilize fast moving neutrons, generating power from plutonium while making more of it from U-238 isotope in or around the fuel, while they get 60 time more energy with original uranium compare to other reactors, they are expensive to build.