There are two ways to release nuclear energy. One is by fusion, where you force two elements to combine - Hydrogen can be used here and Helium is formed. This releases a lot of energy but hasn't yet been used commercially.
The other way is fission, which is used in bombs and nuclear power plants. A heavy element, like Uranium, is forced to split into another element, releasing lots of energy. The splitting of Uranium in this way causes other Uranium atoms to split, so you get a chain reaction.
You're essentially releasing the binding energy of atoms.
Nuclear energy is released by nuclear fission. This happens when large nuclei are split to release energy. A god example of how nuclear fission works is in the mechanism of operation of the atomic bomb and nuclear reactors.