It seems confusing that energy can be generated by both fusion (the coalescence of two nuclei) and fission (splitting the nucleus), as they appear to be quite opposite processes. The explanation lies in the size of the nuclei.
Light elements, such as hydrogen and helium, have small nuclei that release lots of energy when they fuse together. Moving to heavier atoms, less energy is released in each fusion event; until, at iron (26 protons and 30 neutrons), no more energy is released by fusion. Any bigger, it takes energy to make fusion happen. Atoms with really huge nuclei, such as uranium and plutonium do the opposite of fusion: they release energy when they break apart. This is nuclear fission, the process which powers current nuclear power plants.
Fission reactions are much more complex than fusion. For example, uranium-235 can break apart a number of different ways, and many of the atoms produced are unstable and radioactive. This is one attractive thing about fusion, the reaction products are not radioactive: Helium is one of the most stable (and, in a balloon, fun) elements known.