Nuclear power stations are not atomic bombs waiting to go off, and are not prone to "meltdowns".
There is a lot of U-238 in there slowing things down - you need a high concentration of U-235 to make a bomb.
If the reactor gets too hot, the control rods are lowered in and it cools down.
If that doesn't work, there are sets of emergency control rods that automatically drop in and shut the reactor down completely.
With reactors in the UK, the computers will shut the reactor down automatically if things get out of hand (unless engineers intervene within a set time). At Chernobyl, in Ukraine, they did not have such a sophisticated system, indeed they over-rode the automatic systems they did have. When they got it wrong, the reactor overheated, melted and the excessive pressure blew out the containment system before they could stop it. Then, with the coolant gone, there was a serious fire. Many people lost their lives trying to sort out the mess. A quick web search will tell you more about this, including companies who operate tours of the site.
If something does go wrong in a really big way, much of the world could be affected - some radioactive dust (called "fallout") from the Chernobyl accident landed in the UK. That's travelled a long way.
With AGR reactors (the most common type in Britain) there are additional safety systems, such as flooding the reactor with nitrogen and/or water to absorb all the neutrons - although the water option means that reactor can never be restarted.
So should I worry? I think the answer is "so long as things are being done properly, I don't need to worry too much. The bit that does worry me is the small amount of high-level nuclear waste from power stations. Although there's not much of it, it's very, very dangerous and we have no way to deal with it apart from bury it and wait for a few thousand years..
Advantages
Nuclear power costs about the same as coal, so it's not expensive to make.
Does not produce smoke or carbon dioxide, so it does not contribute to the greenhouse effect.
Produces huge amounts of energy from small amounts of fuel.
Produces small amounts of waste.
Nuclear power is reliable.
Disadvantages
Although not much waste is produced, it is very, very dangerous.
It must be sealed up and buried for many thousands of years to allow the radioactivity to die away.
For all that time it must be kept safe from earthquakes, flooding, terrorists and everything else. This is difficult.
Nuclear power is reliable, but a lot of money has to be spent on safety - if it does go wrong, a nuclear accident can be a major disaster.
People are increasingly concerned about this - in the 1990's nuclear power was the fastest-growing source of power in much of the world. In 2005 it was the second slowest-growing.
Is it renewable?
Nuclear energy from Uranium is not renewable.
Once we've dug up all the Earth's uranium and used it,
there isn't any more.
