Although the emergency systems are designed to reinsert the control rods and stop the fission reaction in the event of an emergency, radioactive decay from the reaction products will continue to generate heat in the absence of coolant and fission reactions. This heat will cause the reactor core to melt within an hour after coolant is stopped.
What happens at this point is the subject of conjecture and, perhaps fortunately, little actual experience. The worst case scenario would be if the molten reactor core penetrates the containment vessel and hits ground water. The combination of molten radioactive material and water may cause a chemical explosion which would spread radioactive material over a large area. The best case scenario would be if the containment vessels held the molten material.
Although pressurized water reactors are susceptible to nuclear meltdown in the absence of active safety measures, this is not a universal feature of civilian nuclear reactors, and much of the research in civilian nuclear reactors are for designs with passive safety features that would be much less susceptible to meltdown even if all emergency systems failed.
Fast breeder reactors are more susceptible to meltdown than other reactor types, due to the larger quantity of fissile material and the higher neutron flux inside the reactor core, which makes it relatively more difficult to control the reaction.
A nuclear meltdown is also colloquially known as the China syndrome, from the humorously exaggerated notion that molten reactor material would burrow from the United States through the center of the earth and emerge in China, as popularized by the 1979 film, The China Syndrome.