What if you asked each of your students to write a one or two page explanation of “How a reactor works.”? What would you get? We venture that you would get about as many different explanations as you have students. Disagee? Then try it. The fault is certainly not with the students. It is with the texts and training materials available. Such an overview does not exist ... primarily because it is difficult to integrate a bunch of flawed concepts into any kind of rational explanation. Here’s how it really is.
A nuclear reactor provides an environment conducive to the interaction of a moving neutron and a stationary fissile atom, like U-235. This chance encounter usually produces such severe instability in the atom's nucleus as to split it into two unequal fragments, while releasing two or more (prompt) neutrons, and a tiny amount of energy. If a single prompt neutron from this first fission causes a second fission, and a resultant prompt neutron causes a third fission, etc., then a chain reaction exists. For all operational reactor conditions, each chain is of finite length. The reactor's nuclear status, which determines the number of fission events in a chain, the number of ongoing chains, and whether these numbers are constant or changing with time, is controlled by adjusting the amount of neutron absorber, or poison, within the reactor. The poison, often boron-10, lessens the chance of fission by competing with fissile atoms for free neutrons.
Although prompt neutrons propagate the chain, source neutrons are required to start each chain. In fact, neutron sources, as continuous emitters of neutrons, initiate new chains as existing chains expire. These source neutrons are released by one of two radioactive decay schemes, either from non-fission reactions or fission fragment decay. Non-fission neutrons are a weak neutron source of relatively constant strength that are important in the early stages of reactor startup and the latter stages of reactor shutdown. Delayed neutrons, which are belatedly ejected by a few radioactive fission fragments called precursors, are the principal neutron source because their strength can be increased for attainment of high power.
With many ongoing chain reactions, the neutron population at any given instant consists of both source and prompt neutrons. The prompt neutrons raise the population above the level of the source neutrons alone, a phenomenon known as source multiplication. And, the greater the neutron population, the greater is the power production. In simplest form, the magnitude of reactor power (P) can then be expressed as the product of the total source strength [non-fission (S) plus delayed neutrons (DN)] and of the multiplying effect (M) of the prompt neutrons produced by fissile atoms,or:

Power, itself, reflects the behavior of the chains by exhibiting one of two states, either a steady state where power remains constant with time, or a transient state where power changes with time. A steady state can exist only if both the neutron source and source multiplier are constant with time. Two steady state conditions are encountered operationally, namely equilibrium subcritical multiplication and criticality. At equilibrium multiplication, the reactor power level is sufficiently low that the non-fission source neutrons make up for the prompt neutron losses in excess of precursor production, thereby supporting a steady state. At criticality, the non-fission source is negligible, leaving delayed neutrons as the only neutron source. Prompt neutron losses in successive life cycles are fully compensated by delayed neutrons from precursor decay, resulting in a steady state. The importance of reaching criticality during startup is that, once attained, the delayed neutron source strength can be raised to levels required for high power.
The transient state requires either, or both, the delayed neutron source and source multiplier be changing with time. A transient is always initiated from the steady state by an alteration in the source multiplier (M), causing an incremental change in power. This change in power creates an imbalance between the production and loss of precursor atoms, and the delayed neutron source strength (DN) begins to change accordingly. As the source changes, a further change in power occurs and an ongoing source-power interaction is set in motion. A continuing change in the source multiplier (M), as with ongoing poison adjustment, also produces an ongoing change in power with time.
Reactor power operation is conducted primarily at the steady state condition of criticality, while the transient state facilitates moving from one steady state power to another. When operating around criticality, power has the potential to increase very rapidly but can be reduced only very slowly. This asymmetric response reflects the decay characteristics of the precursor atoms. However, in the Power Range, any mismatch between reactor power and steam demand creates thermal transients which, through negative feedback, tend to return the multiplier (M) toward criticality. This inherent seeking of the steady state is beneficial to reactor control and safety, and tends to mitigate the potential for rapid power excursion.
In summary, a nuclear reactor is nothing more than a multiplier of source neutrons, with reactor power reflecting the magnitude, and behavior, of the neutron source strength and the source multiplier.