Conventional fission power systems rely on a chain reaction of nuclear fission events that release a few neutrons that cause further fission events.  By careful arrangement and the use of various absorber materials the system can be set in a balance of released and absorbed neutrons, known as criticality.
Natural uranium is a mix of several isotopes, mainly a trace amount of U and over 99% U. When they undergo fission, both of these isotopes release fast neutrons with an energy distribution peaking around 1 to 2 MeV. This energy is too low to cause fission in U, which means it cannot sustain a chain reaction. U will undergo fission when struck by neutrons of this energy, so it is possible for U to sustain a chain reaction. There are too few U atoms in natural uranium to sustain a chain reaction, the atoms are spread out too far and the chance a neutron will hit one is too small. Chain reactions are accomplished by concentrating, or enriching, the fuel, increasing the amount of U to produce enriched uranium, while the leftover, now mostly U, is a waste product known as depleted uranium. U will sustain a chain reaction if enriched to about 20% of the fuel mass.
U will undergo fission more easily if the neutrons are of lower energy, the so-called thermal neutrons. Neutrons can be slowed to thermal energies through collisions with a neutron moderator material, the easiest to use are the hydrogen atoms found in water. By placing the fission fuel in water, the probability that the neutrons will cause fission in another U is greatly increased, which means the level of enrichment needed to reach criticality is greatly reduced. This leads to the concept of reactor-grade enriched uranium, with the amount of U increased from just less than 1% in natural ore to between 3 and 5%, depending on the reactor design. This is in contrast to weapons-grade enrichment, which increases to the U to at least 20%, and more commonly, over 90%.
In order to maintain criticality, the fuel has to retain that extra concentration of U. A typical fission reactor burns off enough of the U to cause the reaction to stop over a period on the order of a few months. A combination of burnup of the U along with the creation of neutron absorbers, or poisons, as part of the fission process eventually results in the fuel mass not being able to maintain criticality. This burned up fuel has to be removed and replaced with fresh fuel. The result is nuclear waste that is highly radioactive and filled with long-lived radionuclides that present a safety concern.
The waste contains most of the U it started with, only 1% or so of the energy in the fuel is extracted by the time it reaches the point where it is no longer fissile. One solution to this problem is to reprocess the fuel, which uses chemical processes to separate the U (and other non-poison elements) from the waste, and then mixes the extracted U5 in fresh fuel loads. This reduces the amount of new fuel that needs to be mined and also concentrates the unwanted portions of the waste into a smaller load. Reprocessing is expensive, however, and it has generally been more economical to simply buy fresh fuel from the mine.
Like U, Pu can maintain a chain reaction, so it is a useful reactor fuel. However, Pu is not found in commercially useful amounts in nature. Another possibility is to breed Pu from the U through neutron capture, or various other means. This process only occurs with higher-energy neutrons than would be found in a moderated reactor, so a conventional reactor only produces small amounts of Pu when the neutron is captured within the fuel mass before it is moderated.
More typically, special reactors are used that are designed specifically for the breeding of Pu. The simplest way to achieve this is to further enrich the original U fuel well beyond what is needed for use in a moderated reactor, to the point where the U maintains criticality even with the fast neutrons. The extra fast neutrons escaping the fuel load can then be used to breed fuel in a U assembly surrounding the reactor core, most commonly taken from the stocks of depleted uranium. Pu can also be used for the core, which means once the system is up and running, it can be refuelled using the Pu it creates, with enough left over to feed into other reactors as well.
Extracting the Pu from the U feedstock can be achieved with chemical processing, in the same fashion as normal reprocessing. The difference is that the mass will contain far fewer other elements, particularly some of the highly radioactive fission products found in normal nuclear waste. Unfortunately it is a tendency that breeder reactors in the "free world" (like the SNR-300, the Integral fast reactor) that have been built were demolished before operation, as a "symbol" (as Bill Clinton has stated). The Prototype Fast Breeder Reactor passed tests in 2017 and apparently is about to face the same fate, leaving some military reactors and the Russian BN-800 reactor operating, mostly consuming spent nuclear fuel.