As stated for example in these notes (Link to pdf), top of page 8, irreps of the symmetric group $S_n$ correspond to partitions of $n$. This is justified with the following statement:
Irreps of $S_n$ correspond to partitions of $n$. We've seen that conjugacy classes of $S_n$ are defined by cycle type, and cycle types correspond to partitions. Therefore partitions correspond to conjugacy classes, which correspond to irreps.
I understand the equivalence between partitions, cycle types, and conjugacy classes, but I do not fully get the connection with irreps:
I can associate to a partition $\lambda\vdash n$ the conjugacy class of permutations of the form $$\pi=(a_1,...,a_{\lambda_1})(b_1,...,b_{\lambda_2})\cdots (c_1,...,c_{\lambda_k}).$$
The fact that conjugacy classes are defined by cycle types comes from the fact that $\sigma\pi\sigma^{-1}$ has the same cycle type structure as $\pi$.
However, in what sense do conjugacy classes correspond to irreps? I can understand this if we restrict to one-dimensional representations, as then $\rho(\pi)=\rho(\sigma\pi\sigma^{-1})$ for all $\sigma$, but this is not the case for higher dimensional representations I think, being $S_n$ non-abelian.