Draw a directed graph (loops and multiedges allowed), you get a quiver. Put a vector space at each vertex, a map between vector spaces on each edge, you get a quiver representation.

Quivers are like animals - they can be of finite, tame, or wild type. Saying a quiver is of finite type means that it only has finitely many indecomposable representations. Gabriel's theorem says that this happens if and only if the underlying undirected graph of the quiver is an ADE type Dynkin diagram.

You might be convinced by Sasha some time ago that Dynkin diagrams are very mysterious. I will hopefully demystify them a little bit by sketching a proof of Gabriel's theorem.