There are several critical (pun intended) parts to a nuclear reactor.
Firstly, the off switch. Nuclear reactors have control rods which, strangely enough, control the reaction. They absorb neutrons which could trigger off more reactions which release more neutrons - a chain reaction. When they need to turn the reactor off permamently, they drop the rods in fully absorbing as many neutrons as possible and shutting the reactor down. If the hydraulics don't work, they cut the chains the rods hang from with small explosive devices, letting it be gravity powered. If the explosives fail to work, someone has to climb in with a pair of bolt cutters.
The uranium has to be enriched: a process which involves spinning the uranium around very fast in a centrifuge. This increases the amount of U-235 relative to U-238 (from 1% to 4%) and means that the reaction can take place. However, workers can't keep more than 250 grams in any one place, as it then reaches critical mass and could undergo a chain reaction of its own. This is because the neutrons that are occasionally given off by Uranium as it undergoes radioactive decay will, on average, produce one other fission, which causes it to go off like a bomb.
Another vital part is the moderator. Often graphite or heavy water, the moderator slows the neutrons down. This is not a safety feature, but an important part of the power generation process. If the neutrons are travelling too fast (as they do when they are emitted from the uranium) they won't cause many reactions, making the station inefficient. The moderator slows these neutrons down, letting them collide with other Uranium atoms to create more energy.
Then there's the coolant. This cools down the reactor (it gets pretty hot with all those reactions going on!) and takes the heat from the reactor to the heat exchanger where the heat is transferred to another fluid which has a lower boiling point, and becomes steam. This is used to drive a turbine which generates electricity.
Finally, you've got to dispose of your waste. There are three grades of waste: Low, Medium and High. Low is pumped out into the atmosphere, into the oceans or buried: it's so slightly radioactive that you probably wouldn't detect it from the background radiation. Medium is reasonably dangerous, and needs to be carefully looked after. High is so hugely radioactive that it is actually warm to touch (a Bad Idea) and is used to sterilise medical equipment whilst it's being cooled in cooling ponds, which are industrial size swimming pools. | 
