Why does Iron (Fe) have the most stable nucleus? Which factors (fundamental constants/forces) give this stability?

TL;DR: The way the nucleus is bound together results in competing factors that determine its stability. Some factors favor large nuclei, other factors favor smaller ones. The net result has a sweet spot right at nickel-62, not iron-56 (a common misconception). Let me quickly get this out of the way: http://en.wikipedia.org/wiki/Nickel-62 has the highest average binding energy of any nucleus. It's a common misconception that iron is the "most stable nucleus," as written about in this article: http://adsabs.harvard.edu/abs/1995AmJPh..63..653F. Note that even the Wikipedia figure (and its caption on the file's webpage, incorrectly point to iron-56. The stability of a nucleus is determined by how much http://en.wikipedia.org/wiki/Nuclear_binding_energy it has per nucleon (proton or neutron). Since there are many isotopes that would be perfectly stable in isolation (unless the proton is unstable), we instead will talk about the average binding energy per nucleon. The more binding energy in a nucleus, the harder it is to split it apart. Heavy nuclei (above iron and nickel) tend to decay towards nickel, and light nuclei will fuse (if given the right pressure and temperature) towards iron. So, what determines how much binding energy per nucleon? A few things, and they compete with each other. First, the nucleons are stuck together by what's called the "residual strong force," and a rough analogy is a bunch of styrofoam balls stuck together with velcro. Each ball is stuck together to each of its neighbors, but it doesn't feel any ball outside its immediate range. This would lead us to think that every nucleus has the same binding energy per nucleon... ...except that each nucleon on the surface of the spherical nucleus (usually a good assumption) has fewer neighbors keeping is held down. Since larger nuclei have a smaller surface-area-to-volume ratio, larger nuclei are more stable in this regard. But, there are lots of protons in the nucleus that all have positive charge, and the repulsive force of all of the protons is certainly cumulative. Thus, larger nuclei have a disadvantage in this regard. There are more effects that depend on the energy levels of the neutrons and protons, and that explains some of the features on the lower end of the chart above. Basically, the nucleus has energy levels similar to electron orbitals, and it's energetically favorable to have complete shells. If you add up all of these effects, nickel-62 just happens to have the highest binding energy per nucleon. Iron-56 is a close third place, but it happens to be a common end result when elements are made in the end stages of a star's life and in supernovae. If you'd like more information, check out http://en.wikipedia.org/wiki/Semi-empirical_mass_formula.

Someone with more physics than me will probably give a more technical answer. However, here is a simpler one. In stars hydrogen fuses to helium. The mass of the helium nucleus (2 protons and two neutrons)  is less than the total of four protons (the original hydrogen). The difference is released as energy.      At the other extreme, the mass of uranium 235 is higher than the masses of the two elements resulting from (adding the 2 neutrons released as well) a result of its fission. The difference is released as energy (nuclear power, atomic bomb).      You cannot go downhill releasing energy in both directions (both fision and fission), or else you coul just have an infinite cycle.  So how do both of these processes allow the release of energy?      The element with the lowest amount of mass per nuclear particle is iron. Therefore, elements with more or less Mass than iron can release energy. However, fusion of iron, or breaking ir own requires energy. Therefore it is the most stable.

Ni-62 actually has the record min BE/A, at a whiff under 8.8 MeV per nucleon. Fe-56 has minimum MASS per A, and thus undeserved reputation. It profits from a higher ratio of Z/A than Ni-62, but protons are lighter than neutrons, so not all that mass difference is binding energy. Your trivia for today. http://hyperphysics.phy-astr.gsu.edu/hbase/nucene/nucbin2.html