Yesterday I went to my GR (undergraduate) lecture, and the professor exposed the concept of comoving frame of reference. Well, he said that in this particular frame an observer is a privileged observer. This seems a little bit confusing for me, because, GR says that there's no such thing as a "privileged observer". Is there something I'm missing about my professor's statement that allows it to be true?
Your professor is correct. In the context of the FRW-universe one can define local comoving frames. Observers at rest in such frames are comoving observers, for more see Comoving and proper distances
Why are comoving observers privileged? The reason follows from the cosmological principle according to which the " distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale". The FRW-metric which is a solution of Einstein's field equations is based on the cosmological principle. Thus the privilege of a comoving observer is based on the fact that for him the universe looks the same regardless the direction. In contrast observers who move relative to local comoving observers are not comoving themselves. They will see far away galaxies less redshifted in one direction and more redshifted in the opposite direction compared to comoving observers.
From this it follows that comoving observers are at rest relative to the cosmic microwave background, whereas not comoving observers measure a CMBR dipole anisotropy instead, like we do on earth.
It's essentially a myth that there are no privileged observers in GR. There is indeed a class of privileged observers, namely, the observers that are freely falling. (Of course, since the gravitational field cannot be generically uniform, the class of these privileged observers is a local notion--that is to say that the class of freely falling observers varies from point to point.) This is not really that subtle a point to understand--it might be hard/unpleasant to accept but not hard to see that it is true. Assume that you are in space with no apriori information as to whether you are in a free-fall or not. If there indeed were no distinction between any of the frames then you would not be able to tell as to whether you are in a free fall or not. But, of course, you can tell as to whether you are in free-fall or not by simply throwing a bunch of particles. If they all go on straight lines with uniform velocity for a small enough amount of time in a small amount region of space around you, you are privileged (i.e., in free-fall). If they start changing their speeds or bend then you are not in free-fall. (Compare this to the case of special relativity where all the inertial frames are indeed indistinguishable and therefore there is really no way for a spaceship to tell whether it is in motion or not--because there is no such thing as absolute motion or rest--all the experiments would turn out exactly the same in all the inertial frames.)
Unlike SR whose physical content stems from establishing a true physical equivalence between all inertial frames, the physical content of GR is not in establishing a true equivalence among all frames. Rather, it lies in asserting that the local inertial frames are determined based on the mass-energy-momentum distribution in the universe (and in determining exactly how, namely, through Einstein equations) and in identifying that this force we called gravity arises at a point out of the use of a frame that deviates from the local inertial frame at that point.