Clarifications about up and down converting mixers
I) which is the correspondence between up and down converters circuits and circuits which perform sum and difference of frequencies?


To clarify the concept of a "mixer";

What a mixer ideally is, is just an analog multiplier, which takes two input signals and multiply them together to get the output, this is why the multiplication symbol is used to represent an ideal mixer.

It turns out that if you multiply to sine functions with different frequencies, let's call the frequencies fa and fb, then the result is two new sine functions, one with frequency fa+fb, and one with frequency fa-fb (you also get fb-fa, but one of the latter are going to result in a negative frequency, which is out of the scope of this answer)

In order to get either an up-converter, or a down-converter, you just filter out the signal that you don't want after the mixer. Let's say that you wan't to down-convert, well just add a low pass filter after your mixer, and you are only going to get the difference of the frequencies. You wan't to up-convert? add a high-pass filter after the mixer..

What I have described until now is just how to think of ideal mixers, what they do, and how you can think of the difference between an up-converting mixer, and a down-converting mixer, it is all just to do with the filter you use. However, the ways that up and down converting mixers are made in reality depends completely on the application, typically in RF applications any type of non-linear amplifier can be used, with the filter in its feedback path, and the signals can be added together on the input. This will give a similar result to an actual analog multiplier, and after the filter you still only get the signal that you want.


  II) Exactly what does IF represent? Let's consider for instance a receiver. The signal that arrives at the antenna is the result of an amplitude or frequency modulation through which a signal which contains some informations (for instance an audio signal) is shifted in frequency and carried by a high frequency carrier signal. Then the receiver will have to reconstruct the original information signal. So does IF represent this signal? Or is the down - conversion to IF an intermediate stage?


"IF" Stands for "intermediate frequency", and in most cases this is not the final output that you are looking for.

The IF signal is indeed an intermediate stage, as the name suggests. Let's take the case of an AM modulated signal, after down-conversion you still have an AM modulated carrier, it is just at a lower frequency. The same is true for FM, you just get an FM modulated IF which is at a lower frequency than your original RF signal.