Nuclear fission is the transformation of nuclear energy to thermal energy and electromagnetic radiation.
Fission, for nuclear power generation, is a controlled nuclear reaction that takes place in a specially designed nuclear reactor. A fuel (typically uranium 235) is bombarded with neutrons, causing one heavy atom to split into two. This reaction releases tremendous amounts of energy and more excess neutrons, which them bombard more heavy atoms causing them to split, kicking off a chain reaction.
The fragmented remains of the busted atom are fission products.The diagram below shows the fission reaction for the fuel uranium 235, producing the products barium and krypton, and releasing neutrons.
In the fission process, a tiny amount of mass is "lost." This mass has been converted to energy as described by Einstein's E=mc2, where E = energy, m = mass and c = speed of light. The speed of light is, of course, a very large number. Without a lot of math, you can see that a very small amount of matter converts to a very large amount of energy.
The fission products resulting from a nuclear reaction have an imbalance of neutrons relative to protons, making them unstable. Described as radioactive, these nuclei spontaneously emit electromagnetic rays or streams of particles. This spontaneous loss of energy results in the atoms themselves transforming from one type to another, in a process called nuclear decay. These materials are extremely dangerous to humans and in some cases remain so for over a thousand years.
In fusion, a reaction both opposite of and similar to fission, nuclei are forced together in a process resulting in total less combined mass, yielding tremendous amounts of energy.
Fusion is used with fission in nuclear weapons (hydrogen bombs), but, currently, there are no commercial fusion energy production plants in operation or planned. The insurmountable challenges for a controlled, commercial fusion energy production are getting the fuel (different isotopes of hydrogen) to a high enough temperature and then confining it long enough to get the chain reaction going.
As an interesting aside, the sun itself is a natural fusion reactor – its massive gravitational forces create conditions right for fusion.
For more about fusion, here's an excellent site and summary (and my reference for this section), from the World Nuclear Association [2].
Links
[1] http://www.dummies.com/how-to/content/nuclear-fission-basics.html
[2] http://www.world-nuclear.org/info/inf66.html