What are all representations of the quiver $Q$? Let a quiver be $Q=(Q_0, Q_1, s, t)$, where $Q_0$ is $\\{1, 2\\}$. The quiver has only one arrow: $\alpha: 2 \to 2$. What are all representations of $Q$? Thank you very much. In my original questionthe quiver has only one arrow from 2 to 2, not 1 to 2. But it was edited. I changed the question to the original one.

Presumably you mean representations up to isomorphism, else the question is almost vacuous. You can think about representations of the two components separately; the representations of a single vertex with no arrows are just vector spaces up to isomorphism. For the loop on one vertex, you should (if the field is algebraically closed) think about Jordan normal forms; the classification of endomorphisms of vector spaces is just the classification of square matrices up to similarity (where $A$ and $B$ are similar if there exists an invertible $P$ with $B=P^{-1}AP$). (This is still true if the field is not algebraically closed, but the classification isn't given by Jordan normal form).

Then representations of the quiver are just direct sums of indecomposable representations of each of the two components.