We know that representations of a lie algebra $\mathfrak{g}$ can be studied by looking at the representations of the associative algebra $\mathcal{U}(\mathfrak{g})$, the universal enveloping algebra. Ditto for a finite group $G$ and its group algebra $\mathbb{C}[G]$, and for a locally compact group $G$ and $L^1(G)$.
I can't remember where, but I heard a while back that the theory of Hopf algebras is relevant here. Do Hopf algebras somehow help us study the representations of all the associative algebras above, or something? How are Hopf algebras relevant here?
I don't know anything about Hopf algebras except for their definition.
Universal enveloping algebras $U(\mathfrak{g})$ and group algebras $\mathbb{C}[G]$ are both naturally Hopf algebras. The relevance of the Hopf structure to representation theory is that it tells you how to construct the tensor product of two representations and the dual of a representation. In general, the category of modules over a ring is just an abelian category and has none of this extra structure.