a molecular orbital

In an isolated atom, an electron lives in an atomic orbital centred on one nucleus. Bring two atoms together to form a molecule and the electron no longer belongs to a single atom — its wavefunction spreads over both nuclei at once. A molecular orbital is such a delocalized one-electron state, the natural home for an electron in a molecule, and the picture that explains why some atoms bond and others do not.

The standard construction is LCAO — a linear combination of atomic orbitals — where a molecular orbital is written as a weighted sum of the atomic orbitals it is built from, for example psi = c_A phi_A + c_B phi_B for a diatomic. When the atomic orbitals add in phase, electron density piles up between the nuclei and shields their repulsion: this is a bonding orbital, lower in energy than the separated atoms. When they subtract, out of phase, a node appears between the nuclei and density is pushed to the outside: this is an antibonding orbital (marked with a star, e.g. sigma*), higher in energy. Electrons then fill these molecular orbitals from the bottom up, obeying the Pauli exclusion principle and Hund's rules just as in atoms. The net bond order = (bonding electrons - antibonding electrons)/2 predicts whether and how strongly the molecule binds, and orbitals are labelled by their symmetry about the bond axis (sigma, pi, delta).

Molecular-orbital theory explains bond order, magnetism, and molecular spectra — famously predicting that O2 has two unpaired electrons and is paramagnetic, which the simpler Lewis picture misses. Honest caveats: a molecular orbital is a mean-field, single-electron construct that lives inside the Born-Oppenheimer approximation and neglects electron correlation; it is one of two complementary languages (the other being valence-bond theory), and only the one-electron ion H2+ is exactly solvable — everything larger is an approximation, however good.