Standard Dynkin Diagrams 
The symbols in describing the (Cayley graphs of) Coxeter groups in the catalog are Dynkin diagrams.
Each Coxeter group on n generators r_{1},...,r_{n} can be completely specified by the matrix of orders r_{i}r_{j} of products r_{i}r_{j}.
The collection of these numbers defines an nbyn matrix of natural numbers, called the Coxeter matrix of the group.
Since every generator of a Coxeter group is a reflection, the diagonal entries are all 1's.
Since the order r_{i}r_{j} is equal to the order of r_{j}r_{i}, the matrix is symmetric.
Finally, each offdiagonal entry is, without loss of generality, at least 2; otherwise the corresponding generators are identical, and n1 generators would have sufficed.
Dynkin diagrams are a convenient way of representing Coxeter matrices, and are drawn as follows:

Extended Dynkin Diagrams. 
(Note: this notation is not standard) The polytopes in the catalog are all quotients of the Cayley graph of a Coxeter group (by the vertex symmetry subgroup). For example,
Extended Dynkin diagrams are a convenient way of representing polytopes, and are drawn as follows:
Note that "" and "" are considered the same diagram. Note that the extended Dynkin diagram symbol of a polytope is not always unique, e.g. the 6x8 torus can be variously denoted by "", "", "", or "". Exercise: What are all the extended Dynkin diagrams of the hypercube? (Hint: consult the catalog for some examples.) Technically the vertex symmetry subgroup induces an equivalence relation on vertices of the (right)Cayley graph, which induces a graph homomorphism from the Cayley graph down to a "quotient" graph.
Example: The symmetry group of the square is the 8element dihedral group G = D_{8} = "" with Cayley graph the octagon, say generated by a pair of reflections v,e (refer to Figure 1 below).
The vertex symmetry group of the square is the twoelement reflection group {1,v} about a vertex, generated by v.
The (left)cosets of this set, namely {g{1,v}  g in G} define an equivalence relation on vertices of the Cayley graph.
Identifying/merging all Cayley graph vertices in each class results in the quotient graph the square, i.e., "".
Figure 1. Constructing a quotient graph "" from the Cayley graph of its symmetry group "". 
References. 
