What does it mean by "infinities" when dealing with QFT? I found this PDF online here while browsing Nobel Prize winner contributions, which explains a bit about renormalization (a concept for which Kenneth G. Wilson won the Nobel). 
However I was somewhat confused at the beginning of the text. It mentions this: 
What I'm wondering here is, what does it mean by these "all sorts of infinities" in calculations? Id est, What kind of infinities are there that discovering renormalization was such an impressive feat? 
 A: First I would recommend you to check this video:
https://www.youtube.com/watch?v=hHTWBc14-mk 
The video mentions Renormalization in the context of one-loop Feynman Diagrams. 
In slide 14 (check slide 12 for a picture) of http://www.physics.indiana.edu/~dermisek/QFT_08/qft-II-11-1p.pdf you can check an example of an infinity (a divergent integral), and also the solution to avoid it. Don't mind if you don't understand most of it, just focus on the mechanism, this is renormalization in action. 
This is only possible because we work with gauge theories, which construct interactions based on internal simmetries. Discovering that all gauge theories are renormalizable was a great achievement, check Veltman and t'Hooft wikipedia pages for more info. 
If you don't understand what I meant by "which construct interactions based on internal simmetries" I can give you an example based on what I'm working on. After having a lagrangian for an electron field (check https://en.wikipedia.org/wiki/Dirac_equation) you can derive the lagrangian that describes the interaction between the photon and the electron (https://en.wikipedia.org/wiki/Quantum_electrodynamics) by imposing that the electron field lagrangian is invariant under U(1) transformations (an example of internal symmetry). This is a gauge theory, and every kind of them has infinities that can be avoided :) 
(I'm not an expert (yet :) ), so experts out there feel free to correct me, since this corresponds only to my current understanding of the topic and I would be very happy if I learned even more by providing this answer).
