"Nuclear atom" is not a term that makes sense in this context. I assume that "radioactive atoms" or "radioactive elements" is intended.
The current method by which radioactive elements are used to produce energy is through nuclear fission. This term refers to splitting the nucleus of an atom.
By bombarding an atom with a large, heavy nucleus with neutrons, the atom can be made so unstable that it quickly breaks apart.
Each time an atom splits, two smaller/lighter atoms are produced. However, the mass of the new atoms is slightly less than that of the starting atom. The discrepancy arises because some of the mass is converted to energy, and the amount of energy can be calculated using Einstein's famous E = m c2 equation.
Although the energy released is not particularly useful directly in "as-released" form (essentially fast-moving particles and high-energy radiation), energy can be converted between forms in many ways.
In a typical nuclear reactor, the energy from fission is used to heat water and/or steam which then turns turbines in electric generators, thus producing electrical energy.
The amount of energy capturable is greater than that required as input to make the "nuclear fuel" for these reactors, meaning there is potential for a net gain in usable energy.
In fact, the efficiency of nuclear power generation is greater than that from burning fossil fuels. After determining the processes of fission and converting the energy to a usable form, the problem was primarily one of making the reactors financially reasonable and safe to implement. As these have largely been accomplished, nuclear reactors are now considered viable for power generation (though it remains controversial, primarily because of the waste generated).
For a timeline of the history of fission, see the Wikipedia article here.