Generally, nuclear power plants rely on redundant safety systems, both active and passive, to prevent a meltdown in case of an accident like an earthquake or tsunami. Japan's stricken Fukushima Daiichi nuclear plant is what's called a light water reactor, or specifically a boiling water reactor, because the heat generated in the core of the reactor is used to boil water into steam, powering a turbine to generate electricity.
Immediately after the earthquake, the reactor successfully shut down, meaning that control rods were inserted into the core to disrupt the nuclear reactions directly. However, there's still a lot of heat contained in the core, which is still boiling water and making steam, which raises pressure in the reactor and makes things dangerous. To keep itself cool, the reactor depends on a continuous supply of water, and the problem is that the pumps to supply this water haven't been functioning. This means that the reactor gets hotter, more water turns into steam, and the pressure inside increases (making it more difficult to pump water in), and eventually enough water gets turned into steam that the fuel rods themselves get exposed to air, which can cause them to melt. This may be what is currently happening at Fukushima Daiichi.
The root of the problem at Fukushima Daiichi is that the reactor relies heavily on active safety systems, meaning that the safety systems don't work well (or at all) without things like pumps and generators, which themselves rely on external power. More modern reactors (Fukushima Daiichi was built in 1970) try to incorporate passive safety systems. For example, some reactors suspend their control rods over the core on electromagnets with giant springs behind them, ensuring that the rods will shut the core down the instant power is lost. Other reactors have backup cooling systems that are just giant tanks of water on towers, and explosive valves can be used to pump water into the core using gravity.
Even with passive safety systems, though, accidents can still cause reactors to overheat to the point of meltdown, especially in sustained disaster conditions like those in Japan. The next generation of nuclear reactors, called Gen IV reactors, promise to be significantly safer and more efficient while producing less hazardous waste than the current generation, and one design, called a pebble bed reactor, may even be incapable of having a meltdown at all.
The Pebble Bed:
The core of a PBR contains about 380,000 pebbles, which cycle continuously in and out of the reactor. Every 30 seconds, a pebble drops out of the reactor and is inspected for damage and to make sure it's still got enough fuel left inside. If so, it's put back in the cycle, and if not, it's pulled out and a fresh one is put in its place. On average, a single pebble will cycle through the reactor 10 or 15 times over a few years before being removed.
While a PBR is operating, helium is pumped through the spaces between the pebbles to carry away heat. The helium then flows through a turbine, and that's where the electricity comes from. So far, a PBR isn't that different from a conventional nuclear reactor: you put fuel in, it heats up, and you use that heat to produce electricity. What makes a PBR potentially unique, though, is that because of its design, it's capable of passive, inherent safety that makes a meltdown physically impossible.
Let's just skip directly to the worst-case scenario, like in Japan, where failure of the coolant system caused the reactor to overheat uncontrollably. In terms of what would happen to a pebble bed reactor, this means that there'd no more helium coolant. So, okay, as you might expect, the reactor would start to get really, really hot. As nuclear fuel heats up, the uranium atoms start to move faster, making it harder for them to absorb extra neutrons and split, reducing the reactor's power. This is what's called negative feedback, and while it takes place in all reactors, the low fuel density of the pebbles magnifies it in a PBR. As the PBR continues to heat up, the negative feedback gets stronger and stronger until at about 1600 degrees Celsius, the core stabilizes at an "idle" temperature. This temperature is a solid 400 degrees short of what it would take to cause any damage to the fuel spheres or reactor vessel, which are made of a special kind of super strong graphite.
The upshot of all this is that a pebble bed reactor can have the entirety of its supporting infrastructure power down, blow up, get flooded, get stolen, run out of gas, or otherwise fail, all while the entire staff is on vacation, and the only thing that happens is that the PBR will warm up to its idle temperature and... Stay warm. No meltdowns, no explosions, no radiation leaks. The reactor will just sit there and radiate the heat it produces until you cool it back down or take the fuel out. This scenario was tried once, in a prototype PBR in Germany: they shut off the coolant and removed the control rods and watched, and nothing bad happened. A later inspection of the reactor and fuel pebbles showed no damage.
Of course, it's important to understand that PBRs aren't completely safe, and come with their own risks, including the potential for radioactive dust from pebbles rubbing against each other in the core and the difficulty of managing the circulation of the pebbles themselves. And PBRs still produce radiation, which is always dangerous, along with waste materials, although it's worth mentioning that the waste is already contained inside the pebbles, rendering it much safer, and it's so hard to get out of the pebbles that it's useless as a weapon. But the point is that PBRs seem to be safe in a lot of ways that conventional nuclear reactors definitely aren't.
The first PBR was built in Germany in the mid 60s. As an experimental reactor, it had some design issues, but even so, people working there only received about 1/5 as much radiation as they would if they were working at conventional plant. A follow-up was constructed, but it had some additional design issues and a few minor incidents (mostly related to human error) led to its closure in 1989.
It's definitely true that pebble bed reactors are, at this stage of their development, less familiar to the power industry than more conventional designs. They're also more expensive to construct while having only about 1/30 the power density of other reactors. But China, at least, is optimistic about their potential, and already has one test PBR and is planning on building thirty more in the next ten years, and possibly hundreds more by 2050. Part of the reason that China likes PBRs (besides their safety) is that their high operating temperature can be used to efficiently crack steam into hydrogen, which can be piped off and used as an alternative fuel.
Really, the worst part of the disaster in Japan, as far as the industry goes, is that it's going to make it that much harder to convince the public that nuclear power can be safe, clean, and efficient. To put it in perspective, in 2008 Next Big Future calculated how many people are killed per terawatt-hour of electricity generated. On average, there are 161 fatalities related to energy generation from coal for each one of those terawatt-hours, which comprise a quarter of the energy we use on Earth. 36 people die per TWh of oil energy, which is 40% of our energy use. Nuclear power has a deaths per TWh rate of only 0.04 while producing 6% of our energy, which makes it about ten times safer than solar power once you take into account how many people fall off roofs while installing it, and twice as safe as hydro power.
It's certainly true that nuclear power comes with its own host of issues, from reactor safety all the way down the line to spent fuel storage. It's also true that nuclear accidents are terrible, frightening things. But the fact is that nuclear power is a viable, and even a necessary, alternative to fossil fuels, especially as we start thinking about exploring and colonizing other planets. When we go to Mars for the first time, we're not going to be relying on solar power. We're going to have compact, safe, and clean nuclear power along with us, because that's what makes sense. And it's not just the future: by embracing new technology, we can have the safe and clean nuclear power of tomorrow, today.
There's lots more info on pebble bed reactors from Wikipedia, and you can check your facts at MIT. There's also a detailed discussion of modern PBR safety in a 2009 Nuclear Engineering International article, with links to some PBR criticism as well, and a story on the Chinese PBR from Wired.