Atoms such as Uranium and Plutonium have large numbers of protons and neutrons in their nuclei. This makes the nucleus unstable. A small disturbance to the nucleus such as the addition of an extra neutron causes the nucleus to vibrate and break apart. This is called nuclear fission.

If you were to note the mass of the original nucleus before fission and then add up the masses of all the pieces after fission you would find that the total mass after fission is less than the total mass before fission. This missing mass is turned into large amounts of energy. This is the energy that powers nuclear reactors and atom bombs.

Assume we have an atom of Uranium 235, that is it has 92 protons and 143 neutrons in the nucleus. If we add an extra neutron we create Uranium 236 which is unstable. The nucleus vibrates and splits in two creating two new nuclei of approximately equal mass. In addition three neutrons are also produced. As mentioned before the total mass of the final products is less than the mass of the U235 and the initial neutron resulting in release of energy.

You will notice that the reaction was started with one neutron and three neutrons were produced in the reaction. Each of these resulting neutrons can split another nucleus. One nucleus can split three nuclei which can split nine nuclei which can split 27 and so on. In a very short time large numbers of nuclei become involved releasing huge amounts of energy. This is called a chain reaction.

Depending on how the reaction is set up the reaction can be made to proceed quickly with a rapid, explosive release of energy as in an atom bomb or the reaction can proceed much more slowly as in a nuclear reactor resulting in the energy being released over a longer period of time in a controlled fashion.