Obiwan, I am a nuclear engineer.
I'll try to share some of the knowledge here, briefly (at work now...)
Inside a nuclear reactor there is a lot of heat that comes from nuclear fission. There is a concept of critical mass, "A critical mass is the smallest amount of fissile material needed for a sustained nuclear chain reaction." (Wikipedia, great for copy and paste).
Anyway, nobody really knows what's going to happen. For example, PWR reactors that Jack_Klein mentioned control the critical mass by insertion of graphite rods and water between the fission material rods (fuel rods).
It is possible that due to various processes in the damaged reactor(s) a criticality will be reached, followed by a nuclear explosion, breaking all hell lose because all the radioactive material that is now contained inside the reactor will be spread all over Japan and more.
However, I don't think it will happen - although it is a possibility. The problem now is that even if a reactor is shut down, it still generates heat - quite a lot of it. Cooling systems are in place to cool the reactor, but obviously the 9.0 (it was changed from 8.9 to 9.0) earthquake followed by a tsunami obviously fucked up the cooling systems and the backups. This heat can generate a lot of pressure, and the reactor might explode due to the build-up pressure - spreading a lot of radioactive materials. This is exactly what happened in Chernobyl - the reactor exploded because of the pressure - it wasn't a nuclear explosion.
Furthermore, the rods that were burn out (it doesn't mean there isn't any fuel in them) should be placed in a storage. If you put too many of these together, there is a risk of reaching criticality, so from what I understand, the Japanese put a lot of these together and flooded them with water or some other fluid, to prevent them from reaching criticality, and the container was damaged... So this might be a problem as well.
Anyway, what makes me wonder is the security precautions. Chernobyl happened because of stupid people and because the graphite rods could not be inserted into reactor if diesel generator failed. Since then, the precautions were changed so you don't need electricity to insert the rods - the rods *fall* into reactor. I would think that something along these lines would be thought of for the cooling systems. On the other hand, after all it was a 9.0 earthquake followed by tsunami...
New designs work better, like the pebble bed reactor that was mentioned before... But you can't just replace one nuclear reactor with another, there are a lot of economical and engineering issues involved.