Let's talk physics - and don't roll your eyes, I'll keep it simple. There's nothing hard to understand about the nuclear reactions that result in power generation and atom bombs.
The nucleus of an atom consists of protons and neutrons: the smallest atom, hydrogen, has just one proton; the oxygen atom has eight protons and eight neutrons. Theoretically there is no limit to the size of an atom, but there are practical limits.
Protons have a positive charge, and positive charges repel each other, so if there were not a force holding the protons together, they would fly apart. That force is called the "strong" force, and it only operates over small distances. As the number of protons increases, the strong force has a harder time to keep all those antisocial protons together.
The common form (isotope) of uranium has 92 protons and 146 neutrons. This unwieldy pile of stuff is just at the edge of what makes a stable nucleus. If something comes along to bump the nucleus, it will break, the same way a wine glass perched at the edge of a table will fall and break if you brush it with your finger. And when the uranium nucleus breaks, it can break in a lot of ways---just as the wine glass can break into different pieces. The process of breaking an atom into pieces is called fission, and most of the pieces are radioactive, which is a source of quite a lot of trouble.
Einstein's Equation for the Conservation of Mass and Energy
Energy equals mass times the speed of light squared is the most famous equation in physics. The source of nuclear power, whether for a bomb or for electricity, is the change of mass into energy predicted by Einstein's equation. Because the speed of light is incredibly large, the disappearance of just a tiny amount of mass produces the appearance of a huge amount of energy. When a uranium atom breaks into two smaller atoms, the sum of the weights (mass) of the resulting atoms is less than that of uranium---and the difference appears as energy.
Enter the Neutron
The secret to nuclear energy is the neutron. If you shoot a neutron into a uranium nucleus, the nucleus breaks. And some of the energy released when a uranium atom breaks is in the form of more neutrons, which can break more nuclei. The result is called a self-sustaining chain reaction. That's all there is to it: get a pile of uranium, shoot a neutron at it, and watch energy be produced.
The Rest is Engineering
All right, that isn't all there is to it, but that's all the nuclear physics involved and it's exceedingly simple. All the rest is engineering. First, the uranium isotope with 146 neutrons isn't very good for this; the one with 143 neutrons works much better, and this isotope amounts to only 1 and 1/2 percent of naturally occurring uranium. The separation process is not easy; that's why you hear so much about Iranian efforts to 'purify' uranium.
Then, once you have enough of the right uranium, you have to make the reaction proceed at the right rate. If you don't have enough uranium, the reaction will fizzle out; if you have enough, the reaction may still not be controllable. When you hear about control rods in a reactor, these are simply pieces of material that absorb neutrons. They are inserted into and removed from the uranium pile to slow down or speed up the reaction and make Goldilocks happy.
Meltdown
Even a well-controlled nuclear reaction produces a lot of heat. That is, after all, the point of a power plant---the heat produced is converted into electricity. A cooling system is needed, just as your car needs a cooling system to keep the engine from overheating as it produces the energy of burning petroleum to move your car. If the cooling stops, the car or the power plant will overheat.
There are two basic sources of potential trouble with a nuclear power plant: the cooling system can fail and the core of the reactor can overheat and melt; or the system for venting the radioactive products of fission can fail and radiation can be released into the atmosphere. The former can cause the latter.
Modern nuclear power plants are built with many safeguards against accidents: the cooling system has backups; the whole structure is surrounded by a containment vessel, designed to withstand shocks from the outside, and to keep the harmful substances inside.
Shocks that are considered sufficient to compromise the integrity of a nuclear power plant include: a major earthquake; a large tsunami; a powerful tornado; an impact by a large object such as an asteroid. There is also the possibility of human error, or a confluence of events of very low probability, such as failure of all the cooling systems at once.
Can it Happen in the US?
American nuclear facilities are built to high standards of safety. Effects of the Three-mile-island accident were kept to a minimum by the containment vessel. The Russian plant at Chernobyl was built with less stringent safeguards and the damage was extensive as a large amount of radiation was released.
The Japanese plants are built to very strict standards, equal at least to those of the US. Still the massive quake compromised the safety of several plants. The bad news is that eventually something will happen and there will be an accident at an American plant. The good news is that we're not talking about a nuclear bomb explosion. Some venting of radioactive material into the atmosphere is likely, but the dangers from this can be mitigated by evacuations. Every form of energy production has dangers, and the advisability of using nuclear power has to be judged relative to methods that constantly produce greenhouse gases.