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In a non-atomic or chemical bomb, the energy providing the explosion comes from the breaking of chemical bonds and the joining of others. An example would be ANFO, Ammonium Nitrate / Fuel Oil, probably one of the most used chemical explosives today. During the explosion the nitrate reacts with the fuel oil to produce Nitrogen, CO2 and water along with heat and pressure.
In an atomic bomb the explosive energy is provided by the breaking of unstable elements, such as Uranium 235 which has a natural tendency to break down by the emission of neutrons, this is known as fission. If a large enough amount of the pure fissionable material is in contact, then it can reach critical mass as the neutrons released from one atom's decay cause two more atoms to decay, this proceeds exponentially. During fission a small amount of the mass becomes energy and using Einstein's equation, E=MC2 where E is the energy released, M is the mass converted and C squared is the speed of light squared, even a small amount of mass will release a large amount of energy.
The second type of atomic bomb is a fission bomb. In this bomb 2 atoms of Hydrogen are forced together to make an atom of Helium. This is the same atomic process that powers the sun. The fission reaction is started by a fusion bomb providing the high heat and pressure to cause the atoms to unite. Again when the two atoms unite a small amount of mass is exchanged for energy.
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