In actual quantum electrodynamics, the interaction between two electrons is mediated by virtual photons. So the lowest-order (tree-level) Feynman diagram would have two fermions connected by a photon, like this one: http://tap.iop.org/atoms/particles/536/img_full_47373.gif

The diagram is the same regardless of whether the charges are equal or opposite. That is because a negatively charged electron is just a positively charged positron traveling backward in time, insofar as Feynman diagrams are concerned. Indeed, by arranging the external legs of this diagram this way or that way, you can use it to describe the attraction between an electron and a positron; the repulsion between two electrons or two positrons; or an electron and a positron annihilating and then creating a new electron-positron pair.

A quantum field theory like QED basically provides the rules for constructing valid diagrams, in the form of a list of the fields (or particles) that exist in the theory and their interactions. A vertex represents an interaction: the theory lets you calculate the corresponding vertex function. Internal lines in the diagram are "virtual" particles: they are not detectable, but they mediate the interaction. The theory provides another function (the propagator) associated with internal lines. Ultimately, by appropriately multiplying these together, you can calculate a cross-section or amplitude associated with each diagram.

The observable content of each diagram are the external lines, lines that come in or exit through the edge of the diagram. The theory also tells you what parameters (e.g., energy-momentum, charge, polarization) characterize these external lines. In this particular case, two electrons (or an electron and a positron) come in, two electrons (or an electron and a positron) leave. There are (infinitely) many additional diagrams one can draw with the same set of external lines, but more complicated stuff inside (e.g., the photon can split into an electron and a positron, which would then recombine into a photon. Or, an electron can emit or reabsorb a photon. Or, two electrons can exchange more than one photon. And so on.) However, these diagrams have more and more vertices, and in QED, each vertex brings in a factor of ~1/137, so these diagrams provide progressively smaller contributions to the overall amplitude. Therefore, if you only consider the first, "tree-level" (no closed paths) diagram, you already get a fairly accurate result. This is what it means that the theory is perturbative, meaning that each new level of complexity brings in ever smaller perturbations to the basic result. Not all theories are this easy.

OK, not sure how "intuitive" you find my explanation, but I hope it helps.

Feynman diagrams are a way of representing the quantities that enter in the calculation of the force. So it might be useful to look at how this is calculated in such a setting. A nice presentation is in this paper by Deser: http://arxiv.org/abs/gr-qc/0411026 ``How Special Relativity Determines the Signs of the Nonrelativistic, Coulomb and Newtonian, Forces''

If I recall correctly, some sort of the intuitive explanation you look for can be found in the Feynman's own book "QED: A strange theory of light and matter".

The wave function evolves according to \psi(t) = exp{iHt}\psi(0), where the Hamiltonian H can be split into a free Hamiltonian H_0, and an interaction Hamiltonian H_i containing A: H = H_0 + H_i. For small time steps, the exponential can be expanded to first order as:

(1 + i(H_0+H_i)dt), or (1 + iH_0 dt/2)(1 + iH_i dt)(1 + iH_0 dt/2). In the Feynman diagram language, that is represented by a line (free evolution,) a vertex (interaction,) then a line again.

No alternative to the classical field lines?

One calculates the total energy of the system of two oppositely-charged particles (using the Goldstone, or linked-cluster, expansion -- the particles may be placed inside a box, so as to be limited to a discrete spectrum) held at two different but infinitesimally close relative distances. This gives one the possibility to calculate the derivative of the total energy with respect to the inter-particle distance. Force is the negative of this derivative. Thus, the diagrams arising from the differentiation of the above-mentioned Goldstone diagrams with respect to the relative distance of particles are the sought-after Feynman diagrams. The lowest-order diagram describes the classical electrostatic force, and higher-order diagrams the force arising from the virtual excitations of the electrons on the particles at issue. (I have here for simplicity assumed that the particles at issue are isotropic, otherwise the relationship between force and displacement is determined by a tensor whose calculation requires evaluation of more partial derivatives than one with respect to the relative distance of the particles.)

Well,  just in case the question is still on the table: The paper by Allday includes a very nice and untechnical discussion of that question:

Allday J 1997 The nature of force in particle physics  Phys. Ed. 32(5): 327-332

It comes down to the point: a single diagram does not describe anything which is actually happening. All relevant diagrams need to be considered together...

@Oliver, this is not very intuitive is it? usually we approximate up to tree level, one or two loops etc. the idea of an infinite series gives a computational tool, renormalization a concept etc. but leaves the force quite abstract.     

The Feynman diagrams don't go farther than the Coulomb/Lorentz force.  Then there is quantisation, gauge theory etc. But the concept of "photon exchange" is spurious and is only a view on a very thin slice of the phenomenon. That's similar to looking at a cut potato, whose result depends on the direction of the cut. Feynman had a very peculiar interpretation of quantum mechanics, expressed through the paths integral, but that is only a reformulation of wave mechanics through the superposition principle. The only genuine principle is linearity, which allows to slice anything in a variety of different ways. Only when everything is put together again have we a real picture of the physics.

In classic (and GR-like) way we can see that charges with same sign do radiate photons and accelerate properly. On the other hand charges with opposite signs fall freely and accelerate apparently...

@Esa, can you come again?

@Asher

Of course free fall here isn't just the same as with normal gravity but some fundamental meaning of general properties of vacuum, speculative thinking but you ask intuitive descriptions...

@Esa, i apologize for being so awfully thick. i do not understand your answer and how it relates to my question.

An intuitive level means an analog. I think You need some analog for Feynman diagrams. I gave one without virtual photons gibberish. Sorry if you got nothing.