I know quite a lot about nuclear engineering so here goes:
Not only do they drop the rods to kill off a nuclear chain reaction, there's also the use of
quenching baffles. The main objective is to stop neutron activity as Epsilon mentioned. The core needs to be covered with water. The water is deionised, and also acts as a handy radioactivity shield.
The problem with most of these is design. The reactors in Japan are a very safe design as far as I know
In Ukraine, then part of the Soviet Union, a power surge caused an explosion in one of the plant's reactors, releasing huge doses of radioactive fallout into the air. Two plant workers died within hours, according to the U.S. Nuclear Regulatory Commission; 28 more died in the following months from radiation poisoning. The fallout from Chernobyl was widespread, and the health effects of the disaster are difficult to quantify. A report from the United Nations Scientific Committee on the Effects of Atomic Radiation found that 6,000 individuals who were under the age of 18 in Ukraine, Belarus or Russia at the time of the disaster had by 2006 contracted thyroid cancer, "a substantial fraction" of whom likely contracted the disease due to radiation exposure.
I studied this accident in enormous detail. What happened was as follows: A load-shedding event caused the reactor to go into thermal runaway due to operator lack of skill and error. If you are climbing a hill with your car and you suddenly depress the clutch, what happens? The engine revs into the red suddenly... The same happened here, there was a sudden load-shedding and the reactor went into "over-rev" and became uncontrollable. The engineers at Chernobyl made a critical mistake and loadshed a far too great portion of the load on the reactor during routine maintenance... leading to an open-loop, runaway condition that led to the core running way too hot...
The fallout from Chernobyl was detected as far away as in Germany, Ireland, London