nonbonding orbital
/ non-BOND-ing OR-bih-tal /
Not every electron in a molecule is busy holding atoms together or pulling them apart. Some sit out the bonding game entirely, like a guest at a party who neither helps nor hinders. A nonbonding orbital is a molecular orbital of that kind — its electrons add essentially nothing to the bond, neither strengthening nor weakening it.
There are two common ways a molecular orbital ends up nonbonding. The first: an atomic orbital simply has no partner of the right symmetry or energy to combine with, so it carries over into the molecule almost unchanged in shape and energy. The second: it builds from atomic orbitals on atoms that are not next to each other, so there is no overlap region between adjacent nuclei to pile charge into. Either way, the orbital's energy lands at or very close to the atomic-orbital level it came from, neither pushed down like a bonding orbital nor up like an antibonding one. Many lone pairs that a Lewis structure draws as dots are really electrons in nonbonding molecular orbitals.
Nonbonding orbitals matter because they are often the highest occupied orbitals in a molecule, and so they are where the action is — they are frequently the lone pairs a Lewis acid grabs, or the orbitals that donate when a ligand binds a metal. In water, two of oxygen's lone pairs live in nonbonding-ish molecular orbitals, and those are the electrons that hydrogen-bond to neighbors and that an acid attacks.
In hydrogen fluoride, HF, hydrogen's 1s can combine with only one of fluorine's orbitals. Fluorine's other 2p orbitals point the wrong way and have no symmetry-matched partner, so they stay as nonbonding orbitals — these hold the fluorine lone pairs.
Orbitals with no symmetry match stay nonbonding and often hold the lone pairs.
'Nonbonding' means roughly zero net effect on bond strength, not 'unimportant' — these electrons are frequently the most chemically reactive ones in the molecule.