Nuclear fission is the splitting of an atom into several smaller fragments. This is induced by bombarding an atom of fuel (usually uranium-235 but could also be plutonium-239, uranium-233, and possibly thorium-232) with neutrons. The fissionable atom then captures the neutron and splits (decays) into two smaller atoms (isotopes such as iodine-131, caesium-137 and strontium 90) and two or three neutrons, which go on to split other fissionable nuclei resulting in a chain reaction.
The combined weight of the fission products is less than the weight of the original nucleus and following Einstein's E=mc2 this loss of mass (about 0.1% of the original mass) is converted into the massive energy output of the reaction.
There are many problems with this technique, the most famous and most crucial is the safety aspect. The products of the fission are all highly radioactive and the process itself results in a substantial amount of both beta and gamma radiation. There is no safe, permanent way to dispose of the radioactive products and further safety issues arise in the mining, refining and transportation of the fissionable material. Also the consequences of incompetently operated reactors was made tragically clear with the Chernobyl disaster.