What happens to chemical shifts of protons close to aromatic systems in ¹H-NMR?
The aromaticity found in certain ring systems (e.g. benzene, see Hückel's rule) causes a magnetic field because of the circular motion of the electrons in the ring (see picture).



(Source: Clayden et al. Organic Chemistry 2nd edition, p. 277)

This additional field enforces the magnetic field of the NMR machine inside the ring, while countering the field outside the ring, causing the signals of the outer hydrogens to shift downfield (i.e. to the larger shifts).

Now for your other questions:



  First, what does being near an aromatic system do to protons? If we
  look at the spectra of 1-napthalenemethanol and then
  9-anthracenemethanol, the peak of the two aliphatic protons of the
  benzylic methylene group get shifted way up. Can someone explain why
  adding in that third aromatic ring does this?


Lookig at the difference in shift between the benzylic methylene protons of 1-Naphthalenemethanol and 9-Anthracenemethanol we must conclude that the induced magnetic field of the second is stronger at that position than that of the former, as there is no other appreciable difference. Intuitively this seems to make sense as the field of the 9-Anthracenemethanol is more uniform at and closer to (with respect to the "right" side of the molecule as in the pictures) the discussed proton. Of course this could/should be confirmed by quantum mechanical calculations, but I do not have the tools or the knowledge to do that (if someone has, please feel free to expand this answer!)



(1-Naphthalenemethanol Proton NMR)



(9-Anthracenemethanol Proton NMR)

  I was also under the impression that protons near electronegative
  elements would be deshielded and be shifted higher. However in the
  spectrum of Benzil, the protons para to the carbonyl group are higher
  shifted than the protons meta to the carbonyl. Why is this? Is it to
  do with conjugation of the carbonyl with the aromatic ring? I'm having
  trouble finding a proton nmr in the literature, so there's nothing to
  compare it to.


Yes, conjugating effects and inductive effects can counter each other, but there is no easy general way to say which will win out in a particular situation, although usually it is the conjugating effect.