Essentially it breaks the symmetry between absorption and stimulated emission by using a coherent laser whose frequency closely matches the transition between the upper level and some other auxiliary level, the lasing medium can be coherently prepared in such a way that absorption between the lower and upper levels is reduced or eliminated.
The theoretical mechanism for this process is quantum mechanical interference between two different paths leading to the excited state. The total transition probability of going from the lower level to the upper level is the square of the sum of the probability amplitudes of two separate paths :
The trick here is to choose the atom-laser quantum system in such a way that the separate amplitudes to reach the upper state have opposite phase and thus sum to zero; which implies that the square of this value which is equivalent to the total probability to make a transition from the lower level to the upper level is also zero. Ironically, by creating more paths to state 2, we can reduce the number of atoms that actually get there.
With a zero probability of absorption, the quantum system is transparent to the radiation of the 1-2 transition. Since stimulated emission is not affected, this implies that Lasing Without Inversion (LWI) is now possible which would only require a very small pump to maintain a very small number of atoms in the excited state. Therefore a population inversion is no longer a pre-requisite for laser action.
Although resonance lines are usually among the largest energy transitions between any of the ion's quantum levels they can't be used in an ordinary lasers because of the strong absorption. This leads to a compromise choice of laser transitions among the upper levels which usually have longer wavelengths. In order to obtain shorter wavelengths x-ray lasers, transitions with larger energy differences must be used. Since the transition energies of ionized atoms scales as the square of the ion charge this translates to a higher degree of ionization and to higher pump powers. With LWI, a lower degree of ionization would be required as compared to the more 'traditional' x-ray lasers because the higher energy resonant transitions could be used. These lower charge ions would require less pump power to achieve and since a population inversion is no longer neccessary only a negligible fraction of this power would be required. This means more of the pump power can be allocated to stripping the ions of their outer electrons.
Most x-ray lasers to date use the inefficient collisional excitation pumping scheme. However, if a recombination pumping scheme is adopted the efficiency can be dramatically increased. Therefore the combination of (a) lower ionization, (b) no need for inversion and (c) recombination pumping, promise to make tabletop x-ray phasers a reality. The applications of such an x-ray laser would be limitless.
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