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 eight binary triples (with entries 0 or 1), and for which two such vertices are connected iff the associated triples have Hamming distance 1.
A link for Kuratowski's theorem is http://en.wikipedia.org/wiki/Kuratowski's_theorem