To make electricity, we typically need hot steam. (The steam moves through a turbine and spins its blades, which spins a magnet, thereby moving electrons through wires coiled around the magnet.)
We keep a finite amount of contaminated coolant water flowing past the uranium within a thick, steel, pill shaped shell (see boiling water reactor vessel).
That shell is inside a very very thick containment building, the sole purpose of which is to contain potential accidents.
The containment building is typically inside a bigger outer building.
Cold water flows into the pressure vessel from a coolant system.
The coolant system is typically powered by the same electricity grid that powers your house.
What happens when the power goes out?
There’s always a backup coolant system, usually powered by a nearby gas turbine, diesel generator, or battery.
But, as the secondary coolant system might be weakened by chaos, it’s usually best to shut down the reactor anyway as soon as that sort of thing happens. This is done quickly and mechanically (see control rods).
This is the stage at which most of the affected Japanese reactors are right now.
The core will keep producing heat as it slowly shuts down. At this point, if the cold water doesn’t keep pumping, the water inside gets too hot, causing the pressure to increase. Pressure can be decreased by opening a valve in the pressure vessel. The hot reactor water has bubbles of radioactive gas, which at this point are released into the containment building. These gases include elements such as Hydrogen, which can be very explosive.
The Fukushima I reactor is at this stage right now. Hydrogen fission gasses seem to have collected, ignited, and destroyed the outer building structure, though the containment building appears to be intact. The weak secondary coolant system is being supplemented by cold seawater.
If any of this doesn’t happen fast enough, the fuel can melt. In a worst case scenario, this can complicate control of the reaction and endanger the structural integrity of the pressure vessel. However, the reaction can be controlled with the addition of boron, and even in the unlikely event that the pressure vessel is breached, the containment building will be largely unaffected.