Dynkin diagram
All the rigidity of a root system can be compressed into a tiny picture: one dot per simple root, with the dots joined by edges that record the angles between them. This cartoon, the Dynkin diagram, somehow carries the complete information of a simple Lie algebra — its whole multiplication table is reconstructible from a handful of nodes and lines. It is one of the most efficient encodings in all of mathematics.
Draw one node for each simple root. Between two distinct nodes alpha, beta, draw a number of edges equal to 4(alpha . beta)^2 / ((alpha . alpha)(beta . beta)), which the integrality conditions force to be 0, 1, 2, or 3, corresponding to angles 90, 120, 135, 150 degrees. When the two simple roots have different lengths (a double or triple edge), an arrow points from the long root toward the short one, recording which is which.
The classification theorem is the climax of the theory: connected Dynkin diagrams are exactly the four infinite families A_n (a line of n nodes), B_n and C_n (a line ending in a double edge, arrow opposite ways), D_n (a line forking at one end), plus the five exceptional diagrams E_6, E_7, E_8, F_4, G_2. Over an algebraically closed field of characteristic zero these correspond bijectively to the simple Lie algebras, so this short list classifies them all.
The Dynkin diagram of sl(n+1, C), type A_n, is a single chain of n nodes joined by single edges: o-o-...-o. Type G_2 is just two nodes joined by a triple edge with an arrow, the smallest exceptional case, of rank 2 and dimension 14.
A_n is a plain chain; G_2 is two nodes with a triple edge — the entire algebra in one picture.
Forgetting the arrows gives the Coxeter (or Coxeter-Dynkin) diagram, which sees only angles, not relative lengths; that is why B_n and C_n have the same Coxeter diagram but distinct Dynkin diagrams (and distinct Lie algebras).