I don’t know if I can contribute more content than has already been added, much less make a simple explanation of the question, but I’ll try not to muddy the waters too much in an attempt to render a different, if not qualitatively more coherent, picture.
Before we go into binding energies and gluons and quantum chromodynamics and all that huff, let’s talk about mass first, because I’m guessing that what you think of as “mass” is not what mass really is, at least not on a fundamental scale. Mass–and by this I am referring to invariant mass, a.k.a. rest mass, a.k.a. ‘real’ mass, or also what strict pedants like to call “the only farking type of mass there really is”–is an invariant property of fundamental particles like electrons and quarks. Per particle, this is an absolutely known property that doesn’t go up or down, however much you may shake it or shine a light on it. We make not know where our eletrons are or what they’re about, but by Og we know what their charge, spin, and mass are.
Now, mass is a type of energy, per Einstein’s well known mass equivilence relation, E=mc[sup]2[/sup] (though it was actually Henri Poincaré who first established the relation, though he didn’t realize the full significance of it), but it’s a very special type of energy, bound up in a field that gives it the property of inertia, or resistance to a change in motion. In a sense, you can say that this inertia is conservative, because (apart from interactions with complementary antimatter particles) this invariant mass never changes, though obviously its momentum–the qualitative vectoral measure of its relative inertia, or more simply, how hard it is to slow down or change direction relative to a frame of motion–can change significantly. In other words, mass is a way to store, or at least localize, kinetic energy.
How do you measure momemtum? In basic terms, you let an object hit a known mass at a known velocity and measure the change. Since momentum of the total system is conserved (even if kinetic energy isn’t, in the case of an inelastic collision) you can deduce what the momentum of the original object was. The thing is, there are particles we know to be massless (or at least lacking invariant mass) like the photon, which also display the characteristic of momentum. So, while only some things have mass, everything has momentum. The thing about a photon, however, is that you can’t add to its velocity, or for that matter slow it down. Photons (and other massless particles, as far as we know and theory allows) all move at c, also known (somewhat tautologically) as the speed of light. If you try to slow a photon down, it just gets dimmer/has a lower frequency/longer wavelength. (It’s actually even more complicated than that, but we don’t want to mix with any quantum electrodynamics at this point.)
So a massy system in motion has momentum, but so do energetic systems without (invariant) mass. Make sense so far? In other words, all energy can be measured in terms of its contribution toward momentum. Mass, on the other hand, can only be measured directly in terms of the gravitational field it creates. Since its really, really difficult to measure the gravity field of small objects, we tend to conflate an object’s momentum with its mass. (You try weighing a helium atom on a scale.)
What does all of this have to do with to do with binding energies? Binding energy, as several others have already noted, is negative; that is to say, it is the opposite of potential energy, or more to the point, it represents the realized potential of the ground state of an unbound particle from its combined state. A cat sitting on the kitchen table has a ground state–well, a “table state”–of so-and-so many joules, based upon what it masses and how high the table is. If we arbitrarily make this the unbound state of the cat, then when it is pushed to the floor it is now reduced to a negative “bound” state based upon the mass of the cat and the height of the table. (The cat will let you know just how negative by the dirty look it gives you as it stalks off in anger.) The entire system will suffer an “energy defect”–a loss of potential energy–by the acoustic energy lost from the sound of the cat’s paws hitting the floor (and the yowl the cat makes at being displaced) but the defect is vanishingly small and will be reabsorbed by the air and walls to be converted into randomized thermal energy, so in a closed system the total amount of energy remains constant. The energy defect can also be seen in the now reduced gravitational field; since the field is more localized it appears “less massive”, although the effect is again, imeasurably small. In much more massive systems the adjoining of two masses results in significant loss of potential via gravitational radiation, but the invariant mass of the system is always constant.
Note that the interchange of energy comes about by interplay of two different and opposing forces; in this case, the graviational force which is attractive, and the electrostatic forces that make the cat, table, and floor appear to be solid. It’s not always obvious, but all “reactions” occur from the opposition of two opposed fields. (This is why (missing) dark matter–which only interacts gravitationally and very weakly via the “weak” nuclear force–doesn’t “clump” together the way normal baryonic matter does; it can’t slow down enough to form concentrations.)
We’re talking about gravitational binding energies here, of course (and making some arbitary definitions of ground for the sake of illustration and feline amusement) but a similar effect occurs with the atomic binding energies that come to play in nuclear fusion. In this case, of course, the binding energy is measured as the negative potential of the nuclear force or residual strong force, which wants to pull things together, albeit on a very short scale, versus the electrostatic force of charged nucleons (protons, unless you want to get all crazy and exotic), which try to push the nucleus apart, at least in anything more advanced than isotopes of hydrogen. The balance between these forces is the ground state of the hydrogen nucleus which has less energy than an individual proton and neutron.
This “mass defect”–actually a reduction in momentum–affects the probabilistic quantum behavior of the nucleons, but doesn’t change their invariant mass one whisker. If they were unbound, they’d appear to reign over a much larger area, but collectively they’re more localized, and the amount of energy in the system that is free to go into kinetic energy–rather than vibrational modes or interactions between the nucleons or however you wish to characterize it–is less. Where is the binding energy stored? It’s stored in connections between the nucleons; in QCD terms, it’s the energy transferred by the “virtual” gluons that are exchanged between nucleons that causes them to stick together, and once you prise them apart, the gluons–being incapable of existing in a deconfined, independent state–vanish in a puff of photons or energy applied to the now escaping nucleons, kind of like when an elastic bracelet snaps and sends beads flying all over. Does that sound like voodoo? Well, complain to Murray Gell-Mann; QCD is his baby, and I’ll be damned if I can really figure out what it all means anyway. It’s like a thoroughly rotton onion; the more layers you peel back, the gooeyer and less substantial it gets.
As previously noted, adding more nucleons does not proportionally increase (or rather, decrease) the binding energies; the difference between the two orginal states and the final state is released, in some form, as energy; either photons, or neutrinos, or surplus neutrons with a big bunch of kinetic energy. The total amount of energy released tends to drop as the reactants become more massive (though not strictly so, especially not on the very low end of the periodic table) but once you get up to iron there’s no more energy to be extracted, and you have to push with extra force just to get more energy. High level elements are stable, but only within a vary narrow “valley of stability”, outside of which they tend to decay spontaneously.
I started to write a lot more about fusion, Coulomb barriers, statistical mechanics and Maxwell-Boltzmann distributions, and quantum electrodynamics (which is a theory I can at least get behind and be confidently stupid about) but I realized that this didn’t really contribute anything to the o.p.'s question, the answer to which I’m hoping not to have completely obfuscated and diverged to extensively from at this point.
Stranger