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In quantum field theory, the mass gap is the difference in energy between the vacuum and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle.
Since exact energy eigenstates are infinitely spread out and are therefore usually excluded from a formal mathematical description, a more pedantic definition is that the mass gap is the greatest lower bound of the energy of any state which is orthogonal to the vacuum.
Theories with massless particles, like the photon or a Goldstone boson, have no mass gap. Yang-Mills theory is assumed to have a mass gap, but this has not been proven. To do so is one of the Clay Institute Millennium problems.
The spectrum of a field theory contains an isolated particle at the mass gap if all the states of energy slightly above the gap are single particle states. This is expected to be true of Yang Mills theory, where the mass gap is the mass of the lightest glueball.
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