Spent nuclear fuel, once used in a reactor, still produces heat while waste products within the fuel rods continue to decay. The fuel elements need several years of permanent cooling by circulating water. If the cooling water is not circulated, it will evaporate and without being covered with water, the zirconium cladding around the hot Uranium fuel rods will start to react with surrounding steam and produce hydrogen. The hydrogen may accumulate and explode as happened in unit 1 and 3 at the Japanese plant. Without cooling the cladding will melt and with a bit more heat the Uranium fuel itself will melt, eventually accumulating at the bottom of the pool and further react there.
The first four of the six Fukushima Dai-ichi reactor units are in trouble. There are some 700 fuel rods within the pressure vessel of each reactor core. There are also additional 3450 used fuel elements in the primary cooling ponds above each reactor. These pools are outside the primary containment, but withing the secondary containment, i.e. the outer building wall. The status of these cooling ponds is unknown. As the roof has blown off violently from unit 1 and 3 their cooling ponds may have blown empty of cooling water and are open to the environment. Additionally all the primary cooling ponds in unit 1 to 4 are likely to have no water circulation. The primary cooling ponds in unit 5 and 6 may not have water circulation either.

GE Boiling Water Reactor Mark I
The primary cooling pond at the upper right below the crane.
(more detailed pdf)
There is a common secondary cooling pond at the Fukushima Dai-ichi facility. It contains some 6290 fuel elements and is at ground level. It may well have been damaged when the Tsunami waves ran through and may have no water circulation. Its current status is unknown.
Additional spent fuel, cooled down buts still radioactive, is kept in dry storage casks at the facility. Their status is unknown but with a weight of 100 ton each the casks may have stayed in place when the Tsunami waves ran through. The current total inventory (pdf) at Fukushima Dai-ichi is 1,760 tons of Uranium.
On Saturday the top of the secondary containment of unit 1 blew off in a hydrogen explosion.
On Monday the top of the secondary containment of unit 3 blew off in a hydrogen explosion. This explosion seemed more violent than the one at unit 1 and huge parts of the structure could be seen rising over a hundred yards into the air before violently coming down.
Some seawater cooling has been restored to unit one and three but is not circulated. The cooling method now used is 'feed and bleed', supply water and release steam, which is not viable as a longer term measure.
Unit 2 may have been damaged by parts which came down from the unit 3 explosion. A central pressure release valve stopped functioning. Early Tuesday there was an explosion in unit 2. This one not at the roof level but at the doughnut shaped (torus) suppression water pool at the bottom of the reactor. The suppression pool is part of the primary containment. Escaping steam and a higher level of radiation was observed after the explosion. This lets one assume that the primary containment at unit 2 is now damaged and reliable further cooling at unit 2 may be impossible. There is a high danger that unit 2 may have a serious meltdown.
Unit 4: The reactor itself was shut down when the earth quake and tsunami happened. But the used fuel pool, which is needed to cool spent fuel that earlier had been removed from the core, ran dry after the electricity supply ran out. The used fuel became too hot and produced some hydrogen. There was a fire at unit 4 primary cooling pool late Monday/early Tuesday. The unit has lost a part of its secondary containment.
Early Tuesday radiation of up to 400 millisievert per hour (400,000 microsievert per hour) was observed within unit 4. Short term exposition to 1000 millisievert per hour has immediate negative health effects. Anything above an accumulated 100 millisivert per year(!) is considered to be longterm dangerous. Part of the staff have been evacuated from the site. Normal background radiation is 0.02 microsievert per hour.
As of now unit 1 and 3 have cooled down below boiling temperature and may be regarded as temporary save. Further complications at these units are likely to occure. The status of unit 2 is unknown. Unit 4 is still without reliable cooling.
Some more details at AllThingsNuclear, Pictures of the damaged reactor buildings at ISIS. Digital Globe satellite picture of damaged reactor buildings.
Regular updates are for now available at the IAEA Incident and Emergency Center and the ArmsControlWonk.