This is a general question. The classification of semisimple Lie algebras by using Dynkin diagrams has always amazed me. And these A,B,C,D,E,F,G diagrams seem to appear quite often in the realm of representation theory (of all kinds of things! Lie algebra, Lie group, quivers,etc). My question is (vaguely put), WHY are these diagrams useful? Are they of some more universal (thus more imaginable, more trivial) constructions? They always seem very mysterious for me.

I'm not an expert and would gladly learn about any updates but, as far as I know, your question has no good answer yet. The problem of deeper (universal) origin of the Dynkin diagrams, i.e., of explaining why they show up in many apparently unrelated areas of mathematics (the ADE problem) was posed by V.I. Arnold more than 30 years ago, see here and here, and apparently is still open; at least the 2004 book Arnold's problems lists no solution. Update: you may wish to look also at this question, in particular at the paper mentioned by Thomas Riepe. 

