If a meltdown takes place — essentially the core of the reactor overheating and damaging the fuel rods themselves — it would be the first since the Chernobyl disaster in 1986 and the Three Mile Island incident in 1979.
The Fukushima Daiichi power plant was shut down automatically on March 11 due to the magnitude 8.9 earthquake that hit Japan. The on-site diesel backup generators also shut down about an hour after the event, leaving the reactors without power and thus without the ability to cool down the core. Japanese officials were operating the cooling system via battery power and were flying in batteries by helicopter to keep the temperature regulated, and TEPCO said early March 12 that it has installed a new mobile generator at the plant to enable the cooling system to operate even after the batteries are exhausted.
An unchecked rise in temperature could cause the core to essentially turn into a molten mass that could burn through the reactor vessel. This may lead to a release of an unchecked amount of radiation into the containment building that surrounds the reactor. This building could be breached if enough pressure builds, or, in this case, if the containment building was already breached through the earlier effects of the earthquake.
At the moment, it would appear that Japanese authorities are still trying to contain the reaction inside the reactor. That indicates that the core has not completely melted and that the reaction has not yet gotten out of hand. However, the situation could quickly become uncontrollable and the added water being pumped into the reactor could rapidly evaporate if the temperatures rise too quickly to be cooled off.