A nuclear explosion happens when the energy of a nucleus of a radioactive element is released. As you and I both know, an atom of almost any element is made of electrons, protons, and neutrons, with the latter two particles belonging to the nucleus of the atom. A radioactive element is one that is unstable and consequently will give off energy and/or particles over time. The rate at which a radioactive atom gives off energy can be increased when the atom interacts with other particles, such as neutrons that don't belong to a particular atom.

In the world of nuclear explosions, there are two processes that lead to energy release:
nuclear fission and nuclear fusion. When a free neutron runs into the nucleus of an atom, it may split the nucleus and release energy. This splitting is known as fission. Fusion happens when two hydrogen atoms fuse and form one helium atom and a lot of free energy.

Nuclear explosions take place in a lot of shapes and sizes. The same type of explosion that can destroy cities is the same type of explosion that can power them. The difference is whether these explosions are controlled or not. I'm guessing you're interested in the uncontrolled explosion of a nuclear weapon.

Nuclear weapons are devices that package enriched radioactive elements for violent delivery. Radioactive elements are all around us; you're on the receiving end of them right now! However, they naturally occur in such low concentrations that you need not be frightened. On the other hand, enriched radioactive elements are elements with such a high concentration of radioactive atoms that they are likely to spontaneously react with each other and release a dangerous amount of energy. People have found uranium and plutonium work very well for this purpose.

Nuclear weapons are bombs that trigger chain reactions of enriched radioactive material. A single fission reaction happens in 500 billionths of a second and releases over 200 millions times the original energy of the free neutron that triggered it! The only nuclear bombs ever dropped were fission bombs. Roughly 2 thousandths of a pound of material released the energetic equivalent of 15,000 tons of TNT in the case of the Hiroshima bomb Little Boy and 20,000 tons of TNT in the case of the Nagasaki bomb Fat Man. This amount of energy released in such a short amount of time was enough to kill over 140,000 people and injure almost 250,000. Fusion bombs release even more energy with less material, but have never been used. Let's hope they never are!