Cayley graph
A Cayley graph turns a group into a place you can walk around. Drop one dot for every element of the group. Pick a set of ‘moves,’ the generators; from each dot, draw an arrow to where that move lands you. The result is a roads-and-cities picture: cities are group elements, roads are generators, and any product of generators is a route. The group's algebra becomes the geometry of this network.
Precisely, given a group G and a generating set S, the Cayley graph Cay(G, S) has vertex set G, with an edge from g to gs for each g in G and s in S. When S is symmetric (closed under inverses) one usually treats it as an undirected graph, often with edges labeled by generators. The left multiplication action of G on itself becomes an action by graph automorphisms, so the graph looks the same from every vertex: Cayley graphs are vertex-transitive.
The picture depends on the chosen generating set — different S give different-looking graphs — but for a finitely generated group the large-scale shape is independent of that choice up to quasi-isometry. This is the cornerstone of geometric group theory: properties of the group (growth, ends, hyperbolicity, amenability) become geometric features of the Cayley graph that survive change of generators.
The Cayley graph of Z with S = {1} is the doubly infinite line ... -1 - 0 - 1 - 2 ... . The Cayley graph of Z × Z with the two standard generators is the square grid in the plane, the integer lattice points joined to their nearest neighbors.
Z is a line; Z × Z is a grid — the algebra is literally drawn as geometry.
Choosing the left-multiplication convention (edges g to gs) makes left translation an automorphism; the dual right-multiplication convention is equally common and merely flips a few formulas.