Let’s consider a nuclear reaction with uranium-235 as the fuel. Inside the fuel rods, a neutron with the appropriate energy collides with a uranium-235 atom and is incorporated into this atom’s nucleus. The uranium atom now has an extra neutron and becomes uranium-236. However, uranium-236 is unstable and immediately decays to two smaller atoms—the fission products. Many different fission products are made, such as cesium-133, iodine-135, etc. Wikipedia has a nice entry explaining the fission product yield for uranium-235.
Breaking atomic bonds also releases energy in the form of heat. The purpose of a nuclear power plant is to capture this heat and turn it into electricity. This is analogous to a fossil fuel power plant, where chemical bonds are broken to release heat.
When the uranium-236 decays, extra neutrons (and some other things) are also released. These are called prompt neutrons because they come directly from the fission reaction. (They’re produced promptly.) These neutrons collide with more uranium-235 and the reaction continues. Fission products can also sit around for a while and then decay to produce neutrons, and these are called delayed neutrons. If neutrons are being produced, the fission reaction will continue, and the rate of reaction will be a function of the number of neutrons being produced.
