So my friend asked to see my lens, and the idiot I am, I let him. Before he fully got his hands on it, he touched the rear part of the lens (the mount) he touched the glass and left a thumb print. I quickly got my pen from my camera bag and started cleaning it. I'm quite new to DSLRs and I'm a student so buying a DSLR left a huge dent in my wallet. I'm just really worried if any problems might occur due to my friend touching the rear side of the glass.
The smudge diminishes general picture quality (adds some fuzziness) until you clean it away.
If you get scratches, the same effect happens, just it is then permanent (as you can't clean scratches away).
Probably the worst that can happen is that you scratch the lens while trying to clean it; and that means microscratches, not big visible ones. Be sure to use the right kind of cleaning material (I think those microfiber clothes they give you with new lenses should be good but I don't know for sure). Clearly, no paper towel or such! They do scratch the surface; invisible to your eye, but quality-reducing nonetheless.
What damage could be caused by touching the rear part of a lens?
What could happen is that a bit of grit stuck to a grubby finger could scratch the rear element. But if you don't see any scratches after your recent experience, it's nothing to worry about.
Also, overly aggressive cleaning can wear away lens coatings over time, and the best way to avoid that is to avoid the need to clean the lens in the first place. So avoid touching the front or rear lens elements if you can, but don't freak out if/when it happens.
The functions of lens coatings include reduction of glare and to address chromatic aberration. Anything that is not chemically inert (which is to say pretty much any liquid or solid that can touch the lens) interacts with those coatings on contact.
So, yes, cleaning the smudge promptly is essential to prevent it from becoming permanent in the coating. However, the best approach is to not let anything touch the coated glass at all.
Obviously the front of the lens is similarly coated, which is why it is good practice to keep a UV filter on it. When that filter gets scratched or scuffed, replace it with the spare you keep in back-stock for exactly that occasion rather than going without until you get a new one.
Updating to address concerns in comment on another question
Unless a lens filters out or completely corrects for UV light, chromatic aberration gives it a slightly different focal point than visible light for which the lens is corrected. If the film or sensor is reactive to it, the result is out-of-focus UV degrades the image. A UV Filter with high quality glass is helpful in those conditions.
However, given a lens in which UV is completely corrected or the film/sensor is not sensitive to it then a naked lens will produce the best possible image. I have generally not found consumer- or even most prosumer-grade systems capable of rendering with sufficient resolution that a UV filter degrades the image. Even then all the other factors such as hand-holding, long focal length and shutter speed generally eclipse any degradation from a good quality UV filter. This has especially been so for me when using small format film and almost all modern sensors but that's just one person's experience.