A Dynkin diagram, named after the russian mathematician Eugen B. Dynkin, is a member of a small family of directed graphs originally used as a shorthand to classify and describe the structure of semi-simple Lie algebras. They are increasingly used and generalized for other mathematical objects having similar combinatorial properties.
The Dynkin diagrams are the undirected graphs drawn in this post.
There are four infinite families of type $A_n$, $B_n$, $C_n$, and $D_n$, and five exceptional diagrams $F_4$, $G_2$, $E_6$, $E_7$, and $E_8$. The diagrams of type $A$, $D$, or $E$ are called the simply-laced Dynkin diagrams since they have no double- or triple- edges.
These diagrams classify a few different mathematical objects, one of note being the root systems of semisimple Lie algebras. Coxeter diagrams allow to enumerate and classify related algebraic or geometric objects such as families of generalized uniform polyhedra (by opposition to only regular ones) and many kinds of tilings in finite dimensional spaces.
Most classical references on Lie theory cover them (Jacobson, Humphreys, Knapp, etc). See John Baez's Week 230 for many references and intriguing connections to many branches of mathematics and physics.
