rt.representation theory - Are Dynkin diagrams of some universal construction?

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.