There are a couple of things that all nuclear reactors do that make them behave quite differently than other, more mundane, heat sources. One is just odd, and the other is a little scary.
The first item, the odd one, is called the "Xenon Precluded Start-up".
When a Uranium (or Plutonium) atom fissions, or splits, you end up with two much lighter atoms, called 'fission products' or sometimes 'daughter products'. A fissionable nucleus can split into a myriad of combinations, but some combinations are more likely than others.
The curve below shows the percentage of fission products of U-235 by mass. In nuclear power school, this was called the Mae West curve :) Note that the more likely fission products have two peaks at a mass of about 95 and 135.
This is important, because it turns out that a major neutron absorber (aka "Reactor Poison") is an unstable fission product daughter called Xenon-135. Xenon is the most powerful neutron absorber known. There is an index that describes the likelihood that an atom will absorb a neutron. The number for Uranium-235 is 504.81. For Xenon-135 the number is 2,000,000. Clearly a powerful poison. It will remove neutrons from a reactor core with a vengeance.
When a reactor is run up to full power, Xenon-135 begins to be created and build up. Only 5% of Xenon-135 is produced directly as a fission product. The remaining 95% is produced as the fission product Iodine-135 decays to Xenon-135. After about 50 hours at full power, the Iodine-135 and Xenon-135 have built up to equillibrium; Xenon-135 it is being created as fast from I-135 decay as it is being destroyed by absorbing neutrons and by undergoing decay. Neutron absorption is the major removal mechanism for Xenon-135 during operation, because of the large index of absorption.
Here is where it gets interesting though. Iodine-135 has a 6.7 hour half-life, meaning that in 6.7 hours, half of the I-135 will decay into Xe-135. In 6.7 more hours, half of the remaining I-135 will decay to Xe-135, and so on. Xenon has a *longer* 9.7 hour half-life.
Now suppose you are operating a reactor that has burned up most of its fuel, and it is nearly ready for refueling. You operate this reactor at full power for 50 hours, and in doing so, build up an equilibrium level of Xenon-135. Now for some reason you need to reduce to half power. The neutron population is now only half of what it was at full power, but the Iodine-135 load in the core is very high, because you were just running at full power. This Iodine-135 inventory is rapidly decaying to the neutron poison Xenon-135, only now there is only half the neutron population available to remove it.
What can happen at this point is that the Iodine-135 decays into Xenon-135 and shuts the reactor down. There is nothing that can prevent it, including withdrawing all the control rods. Remember the fuel is not fresh, so exposing more fuel to neutrons will not have as much effect as it would with a new core. This condition is known as "Xenon Precluded Start-up".
Fortunately, however, this situation doesn't last forever. Xenon-135 peaks about 11 hours after power is reduced, and then the loss due to decay of Xenon-135 exceeds the production of Xenon-135 from the decay of Iodine-135. Eventually the Xenon-135 concentration decays away to the point where the aging reactor core can re-start.
Below is a curve showing the build-up of Xe-135 in a recently shutdown reactor core. If the core does not have positive reactivity greater than 0.1 to 0.5∆k/k (depending on power level and time after shutdown) then the reactor will not be able to start again until the Xe-135 decays away.
Xenon Precluded Start-up is only a significant problem for older cores that have high fuel burnup, and don't have much excess reactivity to compensate for such a powerful poison.
The other quirk that nuclear reactors have is... decay heat.
Decay heat is what bit Three Mile Island and Fukushima in the butt, although in different ways.
When a reactor trips, or scrams, the fission process pretty much stops immediately. Neutron population (and fission) dwindles to a millionth of full power over a few minutes, and below the point where heat could be produced in a second or two.
However, in the core of a reactor that has been run at full power, there are a huge number of fission products that are highly unstable, having just been split. They are emitting copious amounts of alpha, beta, gamma, and neutron radiation. Decay heat is the result of all these types of radiation interacting with matter and causing molecular physical movement of atoms (friction), and expressed as heat.
Below is a pellet of Pu-238, glowing red-hot due to decay heat. Pu-238 has an extremely high rate of alpha decay.
Below are estimates of decay heat in a reactor over time, following a trip from full power.
Using this chart, it's pretty easy to understand how decay heat can cause a meltdown if water becomes unavailable to keep the core from getting hot enough to melt the fuel assemblies.
Consider a large Nuclear Power Station. The electrical output will be perhaps 1300 MW. But because the efficiency of a nuclear power plant is only about 33%, the reactor generates about 3900MW of heat to get that 1300 MW of electricity.
Now look at the chart. Four hours after shutdown, our 3900 MW(thermal) reactor core is still generating 39 MW of decay heat!! Ten days after shutdown, the decay heat is about 0.3% of full output or 11.7 Megawatts of heat. That is still enough heat to melt the fuel if there were no way to remove it.
This is what happened at both Three Mile Island and at Fukushima Daichi. At Three Mile Island, there was a steam turbine trip, followed by a reactor trip, and primary coolant pressure surged due to the temperature transient. A relief valve opened to release this excess pressure. Unfortunately the relief valve did not close again when pressure returned to normal.
The operators did not recognize that they were losing coolant, due to a huge mass of unusual alarms. A steam bubble formed in the reactor vessel, and forced primary coolant into the pressurizer vessel. There was no understanding about why the pressurizer filled up with water suddenly, and ther desire was to return the pressurizer level to normal. The emergency reactor water make-up pumps were turned off in an effort drain the pressurizer. Because this cooling water was shut off, the core remained uncovered, and melted down from decay heat. Ironically, had the operators not stopped the emergency water fill pumps which had started automatically, the core would likely have remained undamaged.
With Fukushima, there was a total loss of power following an earthquake-generated tsunami. The earthquake knocked out offsite power, and the tsunami flooded the electrical switchgear room, which was located in the basement. This scenario occurred in three of the four reactors at the site. Additionally, spent fuel pools that required constant circulation for cooling lost power, and those pools began steaming off.
With a complete loss of power, water could not be circulated through the core to remove heat. Operators had to vent steam to prevent an overpressure condition, which threatened to blow the reactor pressure vessel apart. Eventually the reactor ran out of water, and the core melted down from decay heat. Although it is not certain at this time, it is very likely that the molten fuel melted the bottom of the reactor vessels. This would allow hydrogen to escape into the atmosphere and produce the impressive explosions that were so widely televised. Because the primary containment systems were compromised, large amounts of radioactivity were released into the environment, particularly the ocean.
Eventually, out of desperation, seawater was pumped into the primary coolant loops. It is likely that following the earthquake, no possible human action could have prevented the meltdown. The errors made in this case were electrical design and plant siting errors.