https://i.stack.imgur.com/OjEl0.jpg?s=328&g=1

A quiver is a directed graph, i.e. a place to keep a bunch of arrows.
If we put a vector space at each vertex, we can consider all the ways
to put linear transformations at the edges, and get a vector space with
an interesting group action. Theorem [Gabriel '73]: the graphs for which
this action always has finitely many orbits are exactly the ADE
Dynkin diagrams, which in turn correspond to the finite subgroups $H$ of $\text{SL}(2,C)$.
We'll explore this classification in detail.

It turns out to be most interesting to double all the arrows, and bring
in symplectic geometry. Then the moduli spaces of quiver representations
reproduce many interesting spaces, e.g. the Hilbert scheme of n points
in the plane, cotangent bundles to flag manifolds, and the resolution
of $C^2/H$.