A finite graph is planar if and only if it contains no subgraph that is isomorphic to or is a subdivision of $K_5$ or $K_{3,3}$ where $K_5$ is the complete graph of order 5 and $K_{3,3}$ is the complete bipartite graph with 3 vertices in each of the halfs. Wagner's theorem is an equivalent later result.

References

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Kazimierz Kuratowski.
Sur le problème des courbes gauches en topologie.
Fund. Math., 15:271-283, 1930.