Kuratowski's theorem (statement)

Theorem. A graph is planar iff it contains no subgraph that is a subdivision of $K_5$ or $K_{3,3}$.

Equivalent. A graph is planar iff it has no minor isomorphic to $K_5$ or $K_{3,3}$ (Wagner's form).

Conclusion. These are the two fundamental obstructions to planarity.