The classification isn't known (to me anyway), but there is an interesting notion of 'simplicity' for finite graphs:
A graph homomorphism is a map from vertices to vertices such that adjacent vertices remain adjacent. Two graphs are homomorphism-equivalent if there are homomorphisms in both directions. A graph is said to be a 'core' if all endomorphisms are automorphisms. Every finite graph $G$ maps onto a core, and this core is unique up to isomorphism; moreover, the core can be obtained as an induced subgraph that is the image of an idempotent endomorphism of $G$. So some problems in finite graph theory reduce to problems about cores in the same way that some problems in finite group theory reduce to problems about simple groups.
So what are the core graphs? It is easy to see, for instance, that all complete graphs are cores, and that the only bipartite cores are the complete graphs on less than 3 vertices.
I know about this from Peter Cameron, who has done a significant amount of work on the subject.