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The propagator (Feynman propagator) is the amplitude for a particle to travel from one spacetime point to another. Mathematically, it is the time-ordered vacuum expectation value of two field operators, and it equals the Green's function of the classical field equation. Propagators are the internal lines in Feynman diagrams.
The propagator is the most fundamental object in perturbative quantum field theory. For a free scalar field, the Feynman propagator is defined as D_F(x - y) = <0|T{phi(x) phi(y)}|0>, where T denotes time ordering (placing the later-time operator to the left). Physically, it represents the amplitude for a particle to propagate from spacetime point y to point x when x0 > y0, and the amplitude for an antiparticle to propagate from x to y when x0 < y0. The time ordering ensures the correct causal structure.
In momentum space, the Feynman propagator takes the elegant form D_F(p) = i / (p2 - m2 + i epsilon), where epsilon is a positive infinitesimal. The pole at p2 = m2 corresponds to on-shell particles (real particles satisfying the energy-momentum relation). The i epsilon prescription determines the contour of integration in the complex energy plane and encodes the causal boundary conditions: positive-energy modes propagate forward in time, negative-energy modes backward. This single expression contains all the information about free-particle propagation.
The propagator is also the Green's function of the free field equation: applying the Klein-Gordon operator to D_F gives a delta function, (partial2 + m2)D_F(x - y) = -i delta^4(x - y). This means the propagator describes the field's response to a point-like disturbance -- exactly what a Green's function does in classical physics. Each type of field has its own propagator: the Klein-Gordon propagator i/(p2 - m2) for scalars, the Dirac propagator i(gammamu p_mu + m)/(p2 - m2) for spin-1/2 fermions, and the photon propagator -i g_{mu nu}/(k2) (in Feynman gauge) for the electromagnetic field.
In Feynman diagrams, every internal line is a propagator. When two particles scatter, the interaction is mediated by the exchange of virtual particles, and each virtual particle line contributes a factor of the propagator. The full scattering amplitude is built by connecting propagators at interaction vertices (where the coupling constant enters), summing over all possible intermediate states, and integrating over all possible intermediate momenta. The propagator is therefore the building block from which all perturbative predictions in quantum field theory are constructed.