the propagator
If the vertex is the brick of QED, the propagator is the connector that links two bricks together. Whenever a particle is created at one vertex and destroyed at another — say a photon emitted by one electron and absorbed by a second — the propagator is the mathematical object describing that particle's journey in between. It answers the question: given that something starts here, what is the amplitude for it to turn up there?
The key point is that the particle traveling along a propagator is virtual: it lives only for the brief span between the two vertices and need not obey the usual relation between energy, momentum, and mass that a free, observable particle must satisfy. A virtual photon, for instance, can briefly carry an 'impossible' combination of energy and momentum. The propagator's value is largest when the particle is closest to being real and falls off the further it strays — which is why long-range forces, carried by nearly-real massless photons, are strong, while a heavy mediator's propagator is heavily suppressed, making its force short-ranged.
Propagators are how every quantum field theory stitches interactions into complete processes. In QED there are two kinds: the photon propagator for the wavy internal lines and the electron propagator for the straight ones. The exact form of the photon propagator, with its mass set to zero, is what mathematically produces the famous inverse-square Coulomb law when you add up all the virtual-photon exchanges. So the propagator is not an optional extra — it is where the familiar force laws actually come from.
Compare two forces: electromagnetism reaches across a room, while the weak force barely reaches across a proton. The difference is in the propagator — the photon's mass is zero so its propagator stays large at long range, whereas the heavy W boson's propagator collapses almost immediately, choking off the weak force at tiny distances.
A mediator's mass, encoded in its propagator, sets how far its force can reach.
The propagator is a calculational ingredient inside an amplitude, not a measurable path; you never observe the virtual particle itself, only the final particles that come out.