A mass-gap means that aside from the vacuum (totally empty space), the next higher energy state has an energy which is bigger than zero by a finite amount, not by an arbitrarily small amount. This usually means no massless particles, since massless particles can have arbitrarily low energy.
Another way of saying mass-gap which is somewhat more mathematical is that all correlation functions (the statistical versions of quantum fields) are exponentially decaying, so that the field values in imaginary time are independent when you go out far enough away. This is to contrast with a theory with no mass gap, where the correlations go to zero slowly, as a power of the distance between the points.
The mathematical definition is that there exists a positive constant A such that the energy of any state obeys:
$\langle\psi|H|\psi\rangle \ge \langle 0| H |0\rangle + A$
Where $|\psi\rangle$ is any state and $|0\rangle$ is the vacuum state.