The physics term ground state means the lowest possible energy level of something. The term is commonly used in chemistry and spectroscopy for the lowest state of excitation of an electron (i.e., no excitation, or the lowest electron shell). The term is also commonly used for the physics of nuclei, in which nucleons (protons and neutrons) can undergo excitation and relaxation analogous to that of electrons in atoms.
The term zero-point energy (ZPE) means the same thing and is perhaps the more usual term for quantum-mechanical systems other than atoms and molecules. The term vacuum energy is used for the zero-point energy of with a vacuum (region containing no particles), an energy level which is actually not precisely zero, a consequence of the Heisenberg uncertainty principle.
Regarding electrons, the term ground state is sometimes used despite the existence of states at slightly lower levels, i.e., using the term with regard to the larger set of transitions. Smaller transitions termed the fine structure and hyperfine structure may actually result in multiple ground states very close to the same energy level, only one of which fits the definition of lowest energy. This usage undoubtedly came about because with some equipment (lower spectral resolution) and under some circumstances (line broadening), spectral lines associated with these smaller transitions are not observable.