Simple explanation of a particular diagrams of Feynman In relation to this question posed on the website TeX.SE. I am curious to know the use in Physics of green functions about the signs of feynman diagrams with fermionic fields. 
I have not understood the symbolism of circles with dots and arrows or even that strange symbol where the second sum is present.

 A: I will attempt a short answer, or, rather, the shortest possible answer that doesn't lie to you. You should be aware this is a very complicated topic, and there is a lot of math behind what I'm about to say. You shouldn't take anything too literally; I'll just try to give a rough idea of what Feynman diagrams mean.
Also, forget about the dots in the diagrams you found, and about "fermionic fields". Those are additional details, the general idea behind Feynman diagrams is the same.
Thanks to quantum mechanics, nothing is fully determined. This means that when you launch two particles at each other, as they do in particle accelerators, you can't know for sure what will happen when they collide. All you can do is calculate the probabilities for the different possible end results: all kinds of different particles could come out of the collision, with all kinds of possible energies and velocities.
This is done with a formalism called Quantum Field Theory. According to QFT, particles are excitations in their corresponding fields, and they interact through their fields. To calculate the probability for a certain process we first need another number called the probability amplitude, and here comes the problem: the formula for the amplitude is just hopelessly complicated, and basically impossible to calculate by hand$^1$. So what we do is what we always do with a complicated formula: we split it up into an infinite sum of simpler formulas. We can only ever calculate a finite number of them, but usually what happens is that you only need a few terms to get a very accurate answer.
We could just call it a day, since after all we found a way to do the math, but people noticed something: all these simpler formulas share a certain mathematical structure. Each one of them looks as if it was describing a process where the incoming particles transform into others, interact in certain specific ways, and then turn into the outgoing particles. The total probability amplitude would then be the sum of all the possible ways this could happen. We can then draw a little diagram representing each possible "process", and use certain rules to assign a mathematical formula to each diagram.
For example, here is one diagram that could arise in electron-electron scattering$^2$:

The solid lines are electron lines, and the wavy line is a photon; you can see that two solid lines enter the diagram, and two leave it. We interpret this diagram as saying that two electrons interact through a photon. But remember, the diagram really stands for a formula, made up of all the little symbols helpfully written on it. Now, this is not the only diagram for electron-electron scattering. There could be any number of photon lines in the middle, not just one. A photon line could sprout a little solid line loop, which can in turn be attached to more photon lines. Obviously there are infinitely many possible diagrams having two electrons going in and two going out. Each one stands for a formula, a number you can calculate, and the total probability amplitude is the sum of all these numbers.
Be wary of interpreting the diagrams too literally, though. Strictly speaking, nothing so far tells us that the two electrons are actually interchanging a photon. It's just a mnemonic device, a little picture to help us remember complicated formulas. But since it's very useful to pretend like this is actually happening, people tend to get lazy and speak as if the diagram represented actual particle paths through space, instead of just being a convenient way to remember a formula.

$^1$ Computers can do the calculation in certain cases, but it's not very practical or very accurate at the moment, so it's important to be able to do it by hand.
$^2$ Technically according to the usual conventions this looks like positron-positron scattering, or possibly electron-positron.
