In combinatorics there are very simple basic graphs from which a whole lot of theory came. For example the complete graphs $K_5$ and $K_{3,3}$ which alone provide the ground level for any non-planar graph according to Kuratowski's theorem. Another simple graph that gave rise to a huge amount of theory is Petersen's graph, which I like to think as the graph whose vertices are the ten two-element subsets of $\{1,2,3,4,5\}$, and for which two such vertices are connected iff they are disjoint.
A link for Kuratowski's theorem is http://en.wikipedia.org/wiki/Kuratowski's_theorem
