Nuclear fission is a process by which one large, unstable atom can split into two smaller, more stable atoms. It is the means by which most nuclear weapons are built.
If you take an atom of the heavy, unstable uranium, it will tend to, at some point, break down into smaller atoms. Calculating the mass difference, and applying the mass-energy equivalence equation, it is found that a single atom of uranium, after undergoing fission, can produce 200 MeV, 10 million times the power of a chemical transformation such as the burning of TNT.
Fission is usually accomplished by the bombarding of an unstable atom with a neutron. If the atom accepts the neutron, it tends to become even more unstable, and quickly splits, releasing a large amount of energy.
Since each fission produces 2 or 3 more neutrons, a chain reaction can be conceived of, in which a small energy input leads to a non-stop reaction and a massive energy output. This principle can be applied to nuclear reactors, or, if left uncontrolled, a very powerful bomb.
There is however, a critical mass, that is, a sufficient concentration of fissile material must be present before fission can be induced