A look at the curve of binding energy shows that if you break up a heavy nucleus (heavier than iron) into smaller pieces you have more nuclear binding energy per nucleon. Of course, because the strong nuclear force is very strong indeed, nuclear fission of most atomic nuclei requires more energy than the excess nuclear binding energy that is released by the fission. For some unstable radiocative isotopes, however, such as uranium-235 and plutonium-239, only a small amount of energy is required for fission to occur, and large quantities of energy are released.
The first explanation of nuclear fission was given by Lise Meitner and Otto Frisch in 1939, based on experiments performed by Meitner in collaboration with Otto Hahn, who, oddly enough, got sole credit for the discovery by being awarded the 1944 Nobel Prize in Chemistry. The Frisch-Meitner theory uses Niels Bohr's liquid drop model of nuclear structure to explain the phenomenon. Atomic nuclei are thought of as being similar to drops of a liquid, the nucleus being held together by surface tension provided by the strong nuclear force, just as a droplet of water is held together by hydrogen bonds in the water molecules. When the nucleus is externally excited, such as by absorption of a neutron, the extra energy causes the nucleus to deform, but the strong nuclear force tries to make the nucleus return to its former equilibrium shape. The inertia of the nucleons, however, causes the nuclear shape to overshoot equilibrium, causing it to oscillate. The excited nucleus usually radiates away the energy when it does this, and the oscillations die down. However, if the energy supplied is sufficient, the deformation may cause portions of the nucleus to become further apart from each other, far enough that the strong force attracting the protons in the nucleus to each other is no longer that effective in keeping them together, and the electromagnetic repulsion between the positively charged protons becomes stronger. The nucleus then breaks apart.
For U-235, when even a slow thermal neutron is absorbed, it turns into U-236, which is so unstable that it almost instantly explodes in the manner described above into two fission fragments and 2-3 extra neutrons, releasing total energy in the vicinity of 188 MeV. This is a truly astounding amount of energy to be released in a single atomic-scale event; a typical combustion reaction such as the burning of gasoline releases only a few eV per molecule. A similar process occurs in the fission of plutonium and similar elements. For the far more common uranium-238 however, the neutrons to be used must have an energy of at least 1 MeV for fission to occur; neutrons released in nuclear fusion reactions have energies in this range, which is why most high-yield hydrogen bombs have natural uranium casings.
Once it was seen that nuclear fission normally liberates extra neutrons, it became clear that it could be possible to make a self-sustaining reaction if one could ensure that at the neutrons liberated by one fission would in turn cause at least one other fission afterwards, causing a chain reaction. If in a mass of fissionable material the fission doesn't do this, the reactions will die down and eventually stop; this mass is called subcritical. If one fission produces yet another fission on the average, the reaction proceeds in a controlled manner and constant energy is released; such a mass is called critical, and it is the process that powers nuclear reactors. If more than one fission is caused on the average by another fission, that mass is called supercritical, and this generally results in an explosion such as that of an atomic bomb. Fortunately, the conditions for such a runaway nuclear chain reaction to occur are only present in highly enriched uranium or plutonium; even uranium fuel rods are only sufficiently enriched to provide at most a critical sustained chain reaction.