HOMO stands for highest occupied molecular orbital. LUMO stands for lowest unoccupied molecular orbital. The energy difference between the HOMO and LUMO is termed the HOMO-LUMO gap. HOMO and LUMO are sometimes referred to as frontier orbitals.
Each molecular orbital has a calculated energy level. Chemists sort the molecular orbitals (MOs) by energy levels. Chemists assume that the electrons will occupy the lowest energy level MOs first. For example, if a molecule has enough electrons to fill 15 MOs, the 15 MOs with the lowest energy levels will be occupied. The 15th MO on the list would be called the "highest occupied molecular orbital" (HOMO) and the 16th MO on the list would be the "lowest unoccupied molecular orbital" (LUMO). The difference in the HOMO's energy level and the LUMO's energy level is called the band gap. The band gap can sometimes serve as a measure of the excitability of the molecule: the smaller the energy, the more easily a molecule's electrons will be excited. For example, this can help predict whether a substance will have luminescence.