Hi I just heard this phrase from my classmate yesterday, I am wondering whether someone can explain this real quick for me.
In quantum field theory, the propagator is a correlation function between two points in momentum-space. In order to get any useful conclusions from quantum field theory (like, for example, deriving the expectation value of an operator), one must integrate over all possible paths; the propagator between the endpoints of subsections of that path plays a critical role.
One difficulty that can sometimes arise is that the propagator may not necessarily yield a finite integral. One way it can fail to converge is if it has one or more poles (i.e. points where it goes to infinity) at various places along the real line. These poles define quasiparticles. Fortunately, complex analysis allows us to avoid these poles by dipping slightly into the complex plane. Doing so splits the propagator into two independent parts: the regular part, which is the part that behaves nicely with respect to integration, and a quasiparticle propagator which is weighted by the residue of the pole of the original propagator. This residue is what we call the quasiparticle weight; it's the spectral weight of a quasiparticle with a given momentum.
By splitting the propagator in this way, we're able to turn a weakly-interacting theory into a non-interacting one, which is much easier to deal with.
Note that this property is distinct from quasiparticle mass, just as electron spectral weight is different from electron mass.