A nucleus is the center of something. In science, the nucleus can refer to the center of an atom or a cell.
The nucleus of an atom is made of protons and neutrons. Protons are positively charged subatomic particles while neutrons are neutrally charged. It is in the nucleus where all the atomic mass resides (electrons orbit the nucleus of an atom, but the mass of an electron is insignificant to the mass of the entire atom).
The nucleus of a cell is where genetic material is stored (DNA, which stands for deoxyribose nucleic acid). The DNA is what is responsible for passing on traits from one generation. It also codes for the proteins that are made within an organism. The proteins are responsible for the shape, and therefore the function, of most biomolecular structures.