The smallest non-zero Lie algebra of all is the one-dimensional Lie algebra.
If you only look at finite dimensional representations, its irreps are in bijection with pairs $(\lambda,n)$ with $\lambda\in\mathbb C$ and $n\in\mathbb N$: this fact is an immediate consequence of the theorem of Jordan Canonical forms.
If you want to consider infinite dimensional representations, then things are much more complicated. A representation of this algebra is roughly the same as an endomorphism of a vector space, so in a sense the study of its representions is the same thing as operator theory: whole books have been written about this.