Nuclear Fission

If you want a proper explanation, go to Wikipedia. This is an oversimplified version for the sake of the story (translation: word count and my boredom).

Overall: When a neutron hits and is absorbed into a nucleus, various results will happen. What will happen depends a lot on a lot of factors, including but not limited to: energy of the neutron before absorption (aka the speed of the neutron before hitting the nucleus in even more simplified terms), the properties of the nucleus and probability (aka every other factors that I ignore and don't understand).

Whether it splits the nucleus, or the nucleus absorbs the neutron and stabilizes depend on the nucleus and the neutron itself.

Example:

Note: There are many fissile materials that can be used, most with very low half-life but can still be used and definitely bred from fertile materials in the reactor, given sufficient quantity of neutrons flying around and good enough luck.

A thermal neutron (low energy) has a high likelihood of hitting a nucleus (quantum mechanics, not gonna pretend to understand). If the nucleus is U-235, the end result will be U-236 which is likely to split. If the nucleus is U-238 it is going to be absorbed and turns into U-239. Hence the necessity of enriching natural uranium to have higher percentage of U-235.

A fast neutron (high energy) has a lower likelihood of hitting a nucleus. If the nucleus is U-235, the end result will be U-236 which is slightly less likely to split. If the nucleus is U-238 it is going to be absorbed and turns into U-239, which has a chance to split due to the higher energy.

In this specific case: The nucleus will be split into many pieces, usually 2, but more than 2 is possible. Oh and it mays generate more energy than the energy of the neutron as well.

For our use case: The U-236 (and others) splits into uhh, a lot of possibilities for the result here, and several (2-5) fast neutrons. These new neutrons can then go on to cause more fission, creating a chain reaction.

Moderation: As you can guess from the above sections, thermal neutrons can be more useful than fast neutrons, at least when using fuel like U-235. Therefore, moderators are used to slow neutrons down. There are many types like molten salt or high pressure water, to name 2.

Reflector: To increase chances of neutrons colliding with the nucleus, reflector can be used to reflect the neutrons back into the reactor, since neutrons will fly away out of the reactor. Once again, many materials work as reflectors, as well as many related factors like how many neutrons per unit of time, how energetic are the neutrons and how thick is the reflector.

Controller: Simply materials to absorb spare neutrons and control the quantity of neutron to either increase or lower the output of the reactor or just shut the reactor down by ending the chain reaction entirely. Light, aka, normal water can be used, because the single proton of a hydrogen atom (aka protium) is likely to absorb a neutron and turns into deuterium (aka heavy hydrogen) and creating heavy water in the process.

Note: Heavy water is stable and relatively plentiful in nature compared to other isotopes. Due to its greater density, most heavy water is harvest from deposits or a few hundred meters below sea level in the ocean. Both heavy and light water works as moderator, the high pressure is needed to prevent the high temperature water from turning into steam.

Last thing: In certain rare cases, an unstable nucleus can spontaneously undergo fission and produce neutrons which can result in more fission. This is rare enough that it is only an issue in Pu-240 so far, which can cause nuclear warheads to fizzle (aka chain reaction under nonoptimal conditions and yes, the optimal conditions would result in a nuclear explosion). As a result, most nuclear warheads need to have very low percentage of Pu-240, even lower than the weapon grade (enough to not affect chain reactions much, roughly 6%), which is called supergrade at roughly 2%.