The Quasiparticle (cfr L. Landau) energy of an excitation corresponds to the energy required to remove/add an electron to a many-particle system and is observable in Photoemission experiments. From a theoretical point of view, it is the solution of a non-linear and energy-dependent equation, the Quasiparticle equation, in which all the Many-Body interaction effects among the electrons of the system, are accounted by the Self-Energy.
The GW Approximation on the Electronic Self-Energy, first proposed by L. Hedin, consists in the neglection of the Vertex (or more correctly, in the neglection of all the orders beyond the 0th in the perturbative expansion of the Vertex). The Self-Energy, hence, reduces to a simple direct product (in real space) of the Green Function or dressed electron Propagator G and the dynamically screened interaction W:

or diagrammatically:

The first Ab Initio implementations of the GW approximations on simple bulk insulators and semiconductors are due to M. Hybertsen and S.G. Louie, and R.W. Godby, M. Schlüter and L.J. Sham. They showed that a non-selfconsistent G⁰_LDA W^RPA Self-Energy, can already reproduce, within an error of 1%, the removal/addition energy observed in angle-resolved photoemission experiments.
Since then, the validity of the GW approximation has been confirmed by many other calculations. But some doubts have also been raised, since it seems that a fully self-consistent GW Self-Energy does not improve with respect to the first iteration Self-Energy, and it is even worse. Today the mainly accepted idea is that a fully self-consistent Self-Energy requires also some Vertex corrections (tautology). Many justifications of the validity of the first iteration G⁰_LDA W^RPA approximation have been offered so far, but none is completely satisfactory.
A short introduction to the GW Approximation can be found here.