I’m not an expert on nuclear reactors. I taught in the nuclear power program of the US Navy some years ago, meaning I was competent to discuss some aspects of reactors, and specifically the type the navy uses. So I’m also not some random guy in the street. With that disclaimer in mind, there are a few items to mention with regard to the reactor issues in Japan following the earthquake.

This is not another Chernobyl. The reactor design is very different, and the circumstances are different. The Chernobyl accident (link for the more technically inclined) involved an operating reactor that went prompt critical as the result of operational errors, deliberate disabling of certain safeguards as part of a test, and design flaws. This caused a steam explosion and chemical fires as the carbon moderator caught fire.

A closer analogy would be Three Mile Island.

There have been reports of an explosion, but it must be stressed that this was not a nuclear explosion. The reactors have been shut down. It’s not so easy to cause a nuclear explosion in the first place (bombs require a level of expertise), and a shutdown reactor does not have the capability of sustaining the fission reaction. This leaves us with steam pressure buildup or hydrogen as the most likely culprits, i.e. it’s thermodynamics or chemistry, not nuclear physics, which explains the explosion.

The reactor is shut down, so what’s the danger? The products of a fission reaction are typically radioactive, and subsequent decays also release energy. Shutting down the reactor reduces the fission rate by many orders of magnitude, so it’s effectively zero in terms of heat output, but the radioactive fission products still release up to 6-7% of the plant’s power output. The actual value depends on the operating history; the fission products with long half-lives take longer to build up to steady-state values. This value will drop fairly quickly as the short-lived isotopes decay, but it’s still significant — a reactor rated at 1000 MW will still be producing tens of MW of decay heat. The reactors in question at Fukushima Daiichi are rated at 460 or 784 MW (edit 3/15: AFAIK that’s electrical output; if so, the thermal output is ~ 3x higher)

So shutting down does not mean it’s Miller Time? Right. You need to run pumps and do something with the energy, which usually means piping water to a cooling tower, which means you need to run pumps, and those require electricity. It seems silly, at first glance, that a reactor would need a source of power to run it, but the turbines are probably designed to run at the high power output of the reactor and not off of decay heat. So you have an external power line (lost in the quake), local generators (apparently also damaged) and battery backup. Redundant systems. However, it seems that the damage was severe, so the primary and first backup systems are still offline, and if cooling was lost (batteries have a finite lifetime), the water in the core can boil away.

That sounds bad. Yes. As long as the core stays covered with water, things should be fine. But uncovered, the temperature can rise and fuel elements can begin to melt. Hydrogen is produced, which can explode, and boiling water becomes steam, which raises the pressure in the containment vessel. The latter is why the containment vessel would be vented. You would need to replace that water into the system, which also requires pumps. (This what had happened at TMI, though in that case, the cooling pumps were shut off deliberately owing to a flawed procedure)

So this is serious. Nothing here is meant to imply otherwise. But the term “meltdown” (or worse, if preceded by “Chernobyl-like”) raises all sorts of imagery, most of which is inaccurate.

Here are some links from what look to be credible sources. This is a dynamic situation, so there is a shelf-life to the details.
Nuclear Crisis in Japan: What We Know
Factbox: What happens when a reactor loses coolant