Nuclear describes something that has to do with the nucleus — the central part — of an atom. If you're a nuclear physicist, you do a lot of work with that part of the atom.
Nuclear power, which comes from nuclear fission — the energy released when the nucleus of an atom is split — is a controversial source of renewable energy. The word nuclear is sometimes used broadly to refer to the core of something. Your nuclear family is you, your parents, and your siblings — in a sense, the central part of your world, kind of like the nucleus of an atom.