The problem was that nuclear power plants require constant cooling. When the earthquake hit, the automatic shutdown systems kicked in nigh instantly. The problem is that this only reduces the heat output of the reactor core to ~6-10% of its initial heat output, due to residual heat output from unstable fission products decaying, and when the initial heat output was in the gigawatt range that's still a lot of heat. Since you've obviously shut down the reactor at this point, you can't let it power its own cooling systems like it normally does. They had diesel generators which were supposed to power the cooling systems in the event of such an emergency. Unfortunately, the tsunami got to them and completely disabled them (internal combustion engines/electrical systems really, really hate saltwater). There were also some batteries as a secondary backup, but those only lasted a few hours (ideally, you were supposed to fly in portable generators before those ran out). Sadly, that didn't happen, and we ended up in this whole fiasco of desperately trying to prevent the reactor cores from retaining enough heat to transform themselves into radioactive lava.
The intriguing thought is that if the diesel generators had been better protected, this never would have been such a problem.