Chemical energy is stored in the bonds between atoms. There are about 90 different kinds of naturally occuring atoms in the periodic table of the elements, and there are an enormous number of different ways they can bond together into molecules and solids.
Nuclear energy is stored in the bonds between nucleons in the atomic nucleus. There are only two kinds of nucleons - protons and neutrons, and they can bond together in only about 250 stable configurations in the table of isotopes. Essentially all of the ways nucleons can bond together have been explored using particle accelerators, although there is still active research adding a few more very heavy isotopes and refining measurements of excited states of nuclei.
In the US, Brookhaven National Lab produces, collects, and archives data about nuclei and all their properties.
Incidentally, you often see the statement that nuclear energy comes from converting small amounts of mass into energy according to $E=mc^2.$ While this is not false, it is also true of chemical energy or any other form of energy storage. That was Einstein's whole point when he wrote the formula! For example, two hydrogen molecules plus an oxygen molecule weigh slightly more than two water molecules, by exactly the amount of energy released in the chemical reaction. A charged battery will weigh ever so slightly more than a discharged one, and so on. For chemical bonds or energy levels in atoms, the mass difference is just barely on the edge of detectability - roughly parts per billion. Nuclear bonds store of order a million times more energy than chemical bonds, so mass differences between reactants and products are of order parts per thousand. Weighing atomic nuclei (usually with a mass spectrometer) with an accuracy better than a tenth of a percent is therefore a viable technique for measuring nuclear bonding energies, which is doubtless where the misleading idea that $E=mc^2$ has anything to do with nuclear energy arose.
Nuclear fission refers to the splitting of a heavy atomic nucleus into lighter pieces. Heavy nuclei need more and more neutrons to keep them stable. (The attractive forces between nucleons are very strong but have a very short range compared to the electrical repulsive force between protons, so to keep a bigger nucleus stable it needs a larger proportion of uncharged neutrons.) The two smaller pieces each have a smaller ratio of neutrons to protons than the original heavy nucleus, so a few neutrons are left over after each fission reaction. Thus, a single neutron falling into a heavy nucleus (most famously Uranium 235 or Plutonium 239) causes it to fission, producing several more neutrons. The neutrons have to slow down by pinballing off other nuclei, but if there is a big enough lump of fissionable material for them to do so, those second generation neutrons can produce several more fission events, and the number of fissions grows exponentially with the number of generations since the original fission event. This is the famous nuclear chain reaction that leveled Hiroshima and Nagasaki and which is used by all nuclear power plants.
There is no real analog for the nuclear fission chain reaction in ordinary chemistry. The closest thing is probably a laser, where one photon can trigger an excited atom to decay and release a second photon without destroying the original photon, causing a similar exponentially growing cascade of photons. A fission chain reaction has no activation energy (the nucleus actually sucks the neutron in) so the temperature of the material does not matter - it can happen even if the material starts near absolute zero temperature.
Nuclear fusion, on the other hand, has an activation energy which is extremely large by chemical standards, since the reactants have to hit with enough energy to overcome the very large electrical repulsion from their positive charges. However, if the reactants are hot enough that thermal collisions can overcome this barrier, and if the reaction is exothermic, then it should be possible for the energy of the reaction (the product nuclei have more kinetic energy than the reactant nuclei) to heat the reactants even more, and to heat new reactant material to a high enough temperature for it to react. This thermonuclear fusion is analogous to a burning mixture of chemical fuel.
The temperature required for even a tiny thermal tail of especially fast moving reactant nuclei to fuse is of order 5 keV. The unit of temperature here is the kilo-electron-Volt (keV), where one electron-Volt (eV) is 11,600 Kelvin, so this is a whopping 50 million Kelvin. If you can get it hotter, the reactivity goes up. Each different fusion reaction has its own reactivity curve. This reactivity has units of volume per time, so that multiplying it by the density of reactant nuclei produces the rate at which the reactants are being used up. That is, reactivity times density is the probability per unit time that any given nucleus will react. Thus, if you want a significant fraction (one e-th!) of the fuel to react, the reactivity times the density times the confinement time has to be at least one. (The probability to not react in time $t$ is $e^{-\alpha t}$ if $\alpha$ is the probabilty per unit time to react.) Density times confinement time is called the Lawson criterion. The Lawson criterion and the temperature are the main figures of merit for any thermonuclear fusion scheme.
The final slide of the NIF tour presentation shows the reactivities of all of the commonly mentioned fusion reactions as a function of temperature. You can see that the DT reaction $(D+T\rightarrow\alpha+n)$ has about 100 times higher reactivity than the second most reactive possibility, DD (actually two reactions, $D+D\rightarrow ^3\!\!He+n$ and $D+D\rightarrow T+p$). This means to burn a DD plasma, you have to confine it 100 times longer than you need to confine DT at the same density and temperature. Given that magnetic fusion has been trying for 70 years and laser fusion has been trying for 50 years to achieve the confinement time needed for DT fusion, every other fusion reaction is way out of reach and will remain so for a very long time - quite possibly forever.
Fusion power therefore requires that you make tritium (T), since it is half the fuel you need for the DT fusion reaction. You make tritium by bombarding lithium with neutrons. The other half of the fuel, deuterium (D), is the nucleus of about one in every 6000 hydrogen atoms in nature. Water (H2O) is the usual source for manufacturing deuterium, but if you want pure deuterium you need to separate the one (D2O) molecule you will find in every 36 million water molecules, which makes pure deuterium one of the most expensive natural substances in the world. You often see the false claim that seawater fuels fusion reactors - even in places which you would expect to be trustworthy. The truth is that fusion fuel is half lithium and half heavy water. The notion that heavy water alone suffices (or that heavy water is merely seawater) assumes that fusion energy will come from the DD reaction, which burns 100 times slower than the DT reaction we have been struggling to confine for half a century already.
I remember when I first arrived at Livermore Lab in 1980, John Nuckolls - my boss at the time and the inventor of laser fusion - was having lunch with me and the three other new hires. Us four newbies were chattering away about how "practical" fusion reactors would use DD - or even better the proton-boron triple alpha reaction which produces only charged particles. Nuckolls listened to us blither for a few minutes, then smiled and shut us down with the comment, "What's the matter boys? DT isn't hard enough for you?" Considering that a crew of several hundred of us worked our tails off for the next 42 years (and spent about five billion dollars) to barely begin to burn just a tiny bit of DT, Nuckolls obviously knew what he was talking about from the beginning.
Canada has the largest supply of heavy water (D2O) in the world because their nuclear power plant design uses heavy water unlike essentially everyone else. A large fraction of the heavy water supply in the US is tied up in the NIF laser because many of its optics are made from deuterated potasium dihydrogen phosphate (dKDP), a special salt crystal, and the several tons of heavy water required comprise a significant fraction of all the heavy water in the US. When the laser is eventually decomissioned, we have to carefully redissolve those optics in heavy water and give it back - NIF obtained the use of this much extremely expensive deuterium on loan! (Canada declined NIF's request to part with any of their considerably larger supply, incidentally, to give you an idea of just how rare and expensive heavy water is.)
Yes, but using a very different set of nuclear reactions than any proposed fusion power reactor on Earth. The key difference is that essentially all the hydrogen in the Sun is the lightest isotope, so that the only nucleus that can react is just a proton. But the fusion reaction product, the helium nucleus (or alpha particle), has two neutrons in addition to two protons. Where did the neutrons come from? The answer is that the rate determining step in the Sun's fusion reaction is mediated by the weak nuclear force, which can convert protons into neutrons and vice versa, but which is spectacularly weaker than either the strong nuclear force that binds protons and neutrons together in nuclei or the electromagnetic force. This very slow weak reaction in the Sun converts two protons into a proton, a neutron, a positron, and an electron neutrino (look up the famous solar neutrino problem), and its reactivity is so low that it takes billions of years for the Sun to fuse its protons into helium. If the Sun were made of DT or just deuterium (or any of the other fusion reactions proposed for reactors), it would burn all its fuel in a few billionths or millionths of a second, producing a pretty impressive explosion. (Oddly enough, however, it's probably not as big a blast as a supernova, which gets most of its energy from gravitational collapse, making the full $E=mc^2$ available instead of just a fraction of a percent of it.)
By contrast, all of the fusion reactions in hydrogen bombs or any proposed fusion power reactor only involve the strong nuclear force. That is, the number of protons and neutrons are separately conserved, and there aren't any leptons involved (neutrinos, electrons, or positrons).
The keV is actually a measure of energy - the potential energy of a single electron at 1000 Volts electrical potential. When used as a temperature, it means that a typical particle has a random kinetic energy of one keV. (Actually, the rule in statistical physics is "one half keV per degree of freedom on average", which is 3/2 keV for a single electron or ion moving in 3D.) Chemical bonds all break at energies of a few eV; indeed you can rip the valence electrons completely off of atoms with that much energy. At the DT fusion ignition temperature of 5 keV, every material will be a highly ionized plasma.
A plasma behaves like a gas, except it has a very high electrical conductivity. Because of the high conductivity, you can induce large electrical currents in the plasma and attempt to make a "magnetic bottle" to confine the hot plasma for long enough to burn. This is the idea behind magnetic confinement fusion. The problem is that magnetic bottles are always unstable and leaky.
An alternative strategy is the one used in hydrogen bombs (which are powered by thermonuclear fusion): Heat the fusion fuel rapidly enough for it to burn before it has time to blow apart, just like a chemical explosive reacts faster than it can blow apart. This is called inertial confinement fusion. The fact that hydrogen bombs work makes inertial confinement the only demonstrated technique for achieving thermonuclear fusion on Earth. Modeling inertial fusion is in many ways simpler than modeling magnetic fusion, because for the most part you can ignore the electrical and magnetic properties of the fuel and model it as you would any other compressible fluid. (The exception is how a material interacts with laser light, which very much involves the electrical properties of the plasma.)
Imagine a sphere of DT fusion fuel that you've somehow instantaneously heated to 5 keV. A rough estimate of how long it takes to blow apart is the time it takes for a sound wave to propagate from its surface to its center. Assuming a sphere of radius $r$, that is $r/c,$ where $c$ is the speed of sound. Like the reactivity in the Lawson criterion, the speed of sound is mostly a function of temperature. Thus, the density-confinement time product in the Lawson criterion is equivalent to the density-radius product $\rho r$ in inertial confinement fusion. For the DT reaction, the number you need to exceed to burn half the fuel before it blows apart turns out to be about 6 g/cm². In fact, the very rough approximate formula we use in ICF (inertial confinement fusion) taget design is that the fraction of DT fuel that will burn up before disassembly is $\rho r/(\rho r + 6\,\textnormal{g/cm}^2)$. (For any other fusion reaction, you need a $\rho r$ product of at least a few hundred g/cm².)
We usually shoot for $\rho r=3$ g/cm², which is enough to burn up about a third of the fuel according to the formula. The energy content of DT fuel is $3.4\times 10^{11}$ J/g (17.6 MeV/reaction), and at $\rho r=3$ g/cm² you'll burn a third of the fuel, so you'll get a little over $10^{11}$ J/g. The NIF laser was designed for a maximum DT yield of about 20 MJ (or 10 pounds of TNT equivalent), which means the mass of DT fuel at NIF must be about 200 µg. To get $\rho r=3$ g/cm² with 200 µg of fuel, you can work out that you need a DT fuel radius of about 40 µm and a density a bit over 700 g/cc.
That 700 g/cc required DT fuel density is pretty daunting - the densest element is osmium with a density of 22.5 g/cc, and more importantly the density of solid DT ice is only 0.25 g/cc. That is, if you start with solid DT fuel, you need to crush it into a volume 3000 times smaller than its initial volume. If you want to use a lower density, you need to be able to withstand more than a 20 MJ explosion, with the output energy scaling inversely as the square of the fuel density. The laser energy you need to ignite the lower density target also scales inversely as the square of the fuel density - the 2 MJ NIF laser size was chosen because we thought we had a fighting chance of achieving a factor of 3000 fuel compression. If you want a lower fuel compression, you need to build a bigger, more expensive laser to drive it. Note that the scientific breakeven shot produced only 3 MJ yield, not 20, because we haven't yet managed to get the full factor of 3000 fuel compression.
The only convincing way to both compress the fuel to 700 g/cc density and simultaneously heat it to 5 keV temperature is to begin with a hollow spherical shell of DT ice and apply a huge pressure to the outside of it to implode the shell at a very high speed. For example, a DT ice shell 1 mm in radius and 64 µm thick will weigh 200 µg, which is pretty close to the DT fuel dimensions of an actual NIF ignition target. If you can accelerate this shell inward to a speed of about 350 km/s, it will hit center in a few nanoseconds, and the force of that train wreck can be enough to both compress and heat the fuel to thermonuclear ignition. In order to get to 700 g/cc, you have to be very careful to keep the imploding shell at a relatively low temperature while you are accelerating it - a hot shell will be too stiff (a high temperature gas is less compressible than a low temperature gas) and you won't get the required density.
You need to apply several hundred million atmospheres of pressure to the outside of the fuel shell (as you can work out from the numbers in the previous couple of paragraphs). To do that, you surround the DT fuel shell with a shell of ablator material - it was pure carbon in the form of diamond in the NIF shot that ignited. (Actually, it was a thin shell of pure diamond followed by a diamond shell with a fraction of a percent tungsten doping to shield the fuel from x-rays that would otherwise preheat it, followed by an outermost shell of pure diamond.) You can either shoot the lasers at the outside of this ablator, called direct drive, or you can put the capsule inside a little oven you heat with the lasers and let the thermal radiation from the oven drive the ablator. We chose the latter scheme, called indirect drive, for most of the NIF ignition experiments, because even though it is less efficient, you get much more spherically symmetric ablation pressure that way. Since you have to crush that 1 mm fuel shell down to 0.04 mm for it to ignite, keeping everything perfectly symmetric is of paramount importance.
My personal best guess, after spending 40 years working on inertial confinement fusion, is never.
Paradoxically, I'd say the main lesson to be learned from NIF reaching scientific breakeven is that imploding a fuel capsule is every bit as difficult as it sounds: You have to start with a perfectly spherical shell of cryogenic DT ice and implode it with near perfect symmetry by a factor of at least 25 in radius. The slightest imperfections do indeed cause it to fail. This is expensive but - finally - barely possible to do maybe a few times a year with a lot of hard work by several hundred scientists and engineers.
But in order to build a typical 1 GW power plant, you would need to build and shoot roughly a million of these capsules per day (say 10 per second producing 100 MJ each, and not allowing for any inefficiency in converting the explosion energy to electricity). The total world usage of electricity is 3000 GW, which means you need to shoot a billion of these perfect capsules a day to provide even a third of the world's electricity. Even the scaling up of the semiconductor industry in the past six decades to its current level of about a trillion chips a year pales in comparison to what it would take to build and shoot a billion ICF capsules a year. It's just not practical, even before discussing the huge problems with building the laser and reaction chamber.
It's almost as obvious that magnetic confinement fusion will never produce economical electrical power. The ITER test reactor under construction in France will have cost tens of billions of dollars and taken decades to build before it begins operation in the latter half of this decade. It has only a fraction of the systems which would be necessary for an actual fusion power reactor. In particular, it will be the first tests of the extremely difficult lithium blanket necessary to breed tritium fuel for any fusion reactor. Again personally, I think that will turn out to be an extremely difficult and expensive technology.
A single 1 GW DT fusion power reactor of any sort burns at least 50 g of tritium a day, or 20 kg every year. (The US consumes 500 GW of electricity.) The total amount of tritium ever produced in the US (nearly all of it for nuclear weapons use) is about 250 kg. Tritium is extremely radioactive and requires special handling even in microscopic quantities. It is also a "special nuclear material" which is carefully controlled and monitored because of its use in nuclear weapons.
Also, any sort of fusion power plant will produce large quantities of radioactive material in addition to its tritium fuel, created by the intense neutron radiation. The single 3 MJ NIF shot created enough radiation to force the target chamber area to remain evacuated for several days (that is, workers could only enter the area around the target chamber for a few minutes at a time). A 1 GW reactor would shoot a million targets each with 30 times the NIF yield every day, creating a constant radiation level at least a million times larger than this near the reaction chamber. The same would be true of a magnetic fusion reactor. The only advantage of fusion over fission as far as radioactivity is concerned is that by carefully selecting the materials you use to build the reaction chamber, you can arrange for most of the induced radioactivity to consist of relatively short half-life isotopes. During operation, the area near the fusion reactor will be inaccessible due to high radiation levels.
In summary, if you want nuclear power at any price remotely competitive with what you now pay for electricity, your only practical option is fission. Solving its safety and waste problems is almost certainly easier than developing fusion energy. Personally, I think renewable energy is the only long term solution, with fission energy being a possible short term necessity.
Achieving thermonuclear fusion at laboratory scale has become a grand scientific challenge. It is a spectacularly difficult engineering problem which pushes us to and beyond the limits of what is possible. As John F. Kennedy said in a famous speech about the Apollo moon program:
We choose to go to the moon. We choose to go to the moon in this decade and do the other things, not because they are easy, but because they are hard, because that goal will serve to organize and measure the best of our energies and skills, because that challenge is one that we are willing to accept, one we are unwilling to postpone, and one which we intend to win, and the others, too.
Unlike magnetic fusion, laser fusion at the NIF does have an immediate practical benefit in addition to "organizing and measuring the best of our energies and skills". That is, the NIF laser enables us to study the physics of materials at pressures and temperatures important for the operation of nuclear weapons. The ability to ignite thermonuclear fusion reactions with similar burn rates as in weapons provides neutron fluxes which occur nowhere else. This unique capability of NIF is one of the main reasons the US can maintain our nuclear stockpile without doing underground nuclear tests. The US has not set off a nuclear explosion since 1993, and NIF plays an important role in our ability to continue adhering to the comprehensive test ban. Without the experimental tests we can perform at the NIF, we have no way to verify new computer models that take advantage of modern hardware, or to test new theories about things we never understood about old nuclear explosions.