A little bit of history...
In 1932 an English phycist and Nobel Laureate by the name of James Chadwick discovered the neutron. A few years later Enrico Fermi in Rome discovered various elements that, when bombarded with neutrons, formed radioactive elements (meaning that they spontaneously decay). In 1939, German chemists Otto Hahn and Fritz Strassman bombarded uranium salts with neutrons and discovered that barium was produced. Lise Meitner and her nephew Otto Frisch were the first to suggest that a uranium nucleus, having absorbed a neutron could split with the release of energy, into two smaller, and roughly equal parts (barium in the case of Hahn and Fritz). Frisch called the process fission.
The theory behind nuclear fission is quite simple. Particles in atoms (protons, neutrons) always tend to organise themselves such that a minimum ammount of engergy is required to keep them held together by "strong force" interactions. When a relatively large nuclei (uranium) adopts an extra neutron, the new nucleus will tend to be unstable; the energy required to maintain the nucleus becomes much greater and therefore needs to "reconfigure" itself. So the nucleus breaks up into two smaller, stable, nuclei and sheds the extra energy as well as, in the case of uranium, more neutrons.
As seen in the above figure, a very high amount of energy is given off by the fission process.The fact that it takes one neutron to start the reaction where ( in the case of the above reaction) three are rejected, means that a "chain reaction" can occur. This means that once a reaction of this type is started, it would accually grow in intensity. It is for this reason that nuclear fission can be readily used as a very highly effective energy source. In fact fossil fuels don't even seem to come close to measuring up to nuclear energy. Consider an explosion, by ordinary "chemical" means, of a kilo of TNT. Such an explosion will yield 4.2x106 joules of energy whereas an explosion from a nuclear conversion of a kilo of fuel generates on the order of 4.2x1013 joules (a ten million fold increase!).
Because of the vaste amounts of energy that can be obtained by the fission of uranium, the process is readily used produce heat and in turn electricity in "nuclear reactors". Nuclear reactors heat up water by means of controlled nuclear fission and shoots it through a heat exchanger. The hot water exchanges heat through metal tubing which heats up a secondary water source that produces steam to turn a turbine which then produces electricity. A typical reactor can produce about 1000 megawatts of electricity in a year. The question arises that how does the reactor keep the fission chain reaction from going out of control? This is accomplished by using what are know as "control rods" which are made of materials that tend to absorb a lot of the nuclei in the reaction. This will slow down or even stop the fission chain reaction:

Maybe try a small nuclear reactor simulation...
There are many various designs of reactors used throughout the world. A typical reactor is one which uses light water (H2O) as a coolant and modulator, mainly because of its high density, high specific heat and low viscocity properties. These light water reactors are primarily used in the US. The problem with light water reactors is that H2O tends to absorb a large number of neutrons, to the point that the reactors need to use enriched uranium to continue the chain reaction in the reactor. In Canda, the CANDU reactor is used, which make use of heavy water rather than light water. Heavy water doesn't have the same neutron-absorbing properties of ligh water. This allows CANDU reactors to use the naturally occuring uranium.
Canadian Nuclear FAQ's (CANDUs)
There are other benefits to using nuclear power as oppose to fossil fuels besides the obvious high efficiency of the nuclear process. The carbon emissions into the atmosphere, which is always a big topic with fossil fuel burning, are essentially zero in nuclear plants. The use of nuclear fuels also helps lessen (but not solve) the fuel resource problem. But there are other issues to keep in mind when considering the use of nuclear power. Beside the fear of potential core breach as was experience in the 1986 Chernobyl incident, there is also the threat of the biproduct of nuclear fuel "burning": plutonium. Not being a naturally occuring element, plutonium is the most deadly material on the planet. The fact that millions of tons of plutonium are accumilating around the world is a scary thought indeed. Even scarier is the 37 000, plutonium made, nuclear weapons that still exist on this planet. There is no known way of neutralizing plutonium so governments have opted to simply stock it. This raises other concerns such as that of the transportation of the wast product from nuclear plants.
FOE Alarmed at flight transport of Nuclear waste from UK