Following the events at the Fukushima Daiichi nuclear reactors in Japan has been challenging. At best, even those present at the site have a limited view of what's going on inside the reactors themselves, and the situation has changed rapidly over the last several days. Meanwhile, the terminology involved is somewhat confusing—some fuel rods have almost certainly melted, but we have not seen a meltdown; radioactive material has been released from the reactors, but the radioactive fuel currently remains contained.
Over time, the situation has become a bit less confused, as cooler heads have explained more about the reactor and the events that have occurred within it. What we'll attempt to do here is aggregate the most reliable information we can find, using material provided by multiple credible sources. We've attempted to confirm some of this information with groups like the Nuclear Regulatory Commission and the Department of Energy but, so far, these organizations are not making their staff available to talk to the press.
What goes on inside a nuclear reactor
Nuclear reactors are powered by the fission of a radioactive element, typically uranium. There are a number of products of this reaction, but the one that produces the power is heat, which the fission process gives off in abundance. There are different ways to extract electricity from that heat, but the most common way of doing so shares some features with the first steam engines: use it to boil water, and use the resulting pressure to drive a generator.
Radioactivity makes things both simpler and more complex. On the simpler side, fission will readily occur underwater, so it's easy to transfer the heat to water simply by dunking the nuclear fuel directly into it.
Unfortunately, the radioactivity complicates things. Even though the fuel is sealed into rods, it's inevitable that this water will pick up some radioactive isotopes. As a result, you can't just do whatever you'd like with the liquid that's been exposed to the fuel rods. Instead, the rods and water remain sealed in a high-pressure container and linked pipes, with the hot water or steam circulated out to drive machinery, but then reinjected back into the core after it has cooled, keeping a closed cycle.
The water recirculation doesn't just let us get power out of the reactor; it's essential to keeping the reactor core cool. Unless the heat of decay is carried away from the core, its temperature will rise rapidly, and the fuel and its structural support will melt.
The fission reaction
On its own, the uranium isotope used in nuclear reactors will decay slowly, releasing a minimal amount of heat. However, one of the decay products is a neutron, which can strike another atom and induce that to split; other neutrons are produced as the products of that split decay themselves. At high enough densities, this chain reaction of neutron-induced fission can produce a nuclear explosion. In a nuclear reactor, the fuel density is low enough that this isn't a threat, and the rate of the fission can be controlled by inserting or removing rods of a material that absorbs neutrons, typically boron.
Completely inserting control rods to limit uranium's fission, however, doesn't affect what's happened to the products of previous reactions. Many of the elements that are produced following uranium's split are themselves radioactive, and will decay without needing any encouragement from a neutron. Some of the neutrons from the reactor will also be absorbed by atoms in the equipment or cooling water, converting those to radioactive isotopes. Most of this additional radioactive material decays within the span of a few days, so it's not a long-term issue. But it ensures that, even after a reactor is shut down by control rods, there's enough radioactive decay around to keep things hot for a while.
All of which makes the continued operation of the plant's cooling system essential. Unfortunately, cooling system failures have struck several of the reactors at Fukushima Daiichi.
Surviving the earthquake, but not the tsunami
Because cooling is so essential to a plant's operation, there are a few layers of backups to keep the pumps running. For starters, even if the reactors themselves are taken offline, the coolant pumps can receive power from offsite; this option was eliminated by the earthquake itself, which apparently cut off the external power to Fukushima. The earthquake also triggered a shutdown of the reactors, removing the obvious local source of power to the pumps. At this point, the first backup system kicked in: a set of on-site generators that burn fossil fuels to keep the equipment running.
Those generators lasted only a short while before the tsunami arrived and swamped them, flooding parts of the plant's electrical system in the process. Batteries are in place to allow a short-term backup for these generators; it's not clear whether these failed due to the problems with the electrical system, or were simply drained. In any case, additional generators were slow to arrive due to the widespread destruction, and didn't manage to get the pumps running again when they did.
As a result, the plants have been operating without a cooling system since shortly after the earthquake. Even though the primary uranium reaction was shut down promptly, the reactor cores have continued to heat up due to secondary decay products.