What people mean when they say "meltdown" can refer to several different things, all likely coming after a hydrogen explosion. A "full meltdown" has a more generally accepted definition than, say, a "partial meltdown." A full meltdown is a worst-case scenario: The zirconium alloy fuel rods and the fuel itself, along with whatever machinery is left in the nuclear core, will melt into a lava-like material known as corium. Corium is deeply nasty stuff, capable of burning right through the concrete containment vessel thanks to its prodigious heat and chemical force, and when all that supercharged nuclear matter gets together, it can actually restart the fission process, except at a totally uncontrollable rate. A breach of the containment vessel could lead to the release of all the awful radioactive junk the containment vessel was built to contain in the first place, which could lead to your basic Chernobyl-style destruction.
The problem with a full meltdown is that it's usually the end result of a whole boatload of other chaos--explosions, fires, general destruction. Even at Chernobyl, which (unbelievably, in retrospect) had no containment building at all, the damage was caused mostly by the destruction of the plant by explosion and a graphite fire which allowed the corium to escape to the outside world, not the physical melting of the nuclear core.
Over the weekend, Chief Cabinet Secretary Yukio Edano somewhat hesitatingly confirmed a "partial" meltdown. What does that mean? Nobody knows! The New York Times notes that a "partial" meltdown doesn't actually need to have any melting involved to qualify it as such--it could simply mean the fuel rods have been un-cooled long enough to corrode and crack, which given the hydrogen explosion, we know has already happened. But we'd advise against putting too much stock in any term relating to "meltdown"--it'll be much more informative to find out what's actually going on, rather than relying on a vague blanket term.
As TEPCO grapples with the damage the earthquake and tsunami did to the nuclear system, there's going to be lots of news--there could be more explosions, mass evacuations, and more "meltdowns" of one kind or another. All we can do is learn about what's going on, think calmly about the situation, and hope that TEPCO can keep the damage done by the plants as low as it is now.