Enumerate all non-isomorphic graphs of a certain size
Probably the easiest way to enumerate all non-isomorphic graphs for small vertex counts is to download them from Brendan McKay's collection. The enumeration algorithm is described in paper of McKay's [1] and works by extending non-isomorphs of size n-1 in all possible ways and checking to see if the new vertex was canonical. It's implemented as geng in McKay's graph isomorphism checker nauty.

[1]: B. D. McKay, Applications of a technique for labelled enumeration, Congressus Numerantium, 40 (1983) 207-221.