In addition to the construction of (generalized) Eilenberg–MacLane spaces mentioned in the comments, there are many other examples.

For example, the Dold–Kan correspondence allows us an easy perspective on cohomology operations on simplicial cochains.
If $X$ is a simplicial set and $A$ is an abelian group, then
$$\def\C{{\sf C}}\def\N{{\sf N}}\def\Z{{\bf Z}}\def\Hom{\mathop{\sf Hom}}\C^*(X,A)=\Hom(\N\Z[X],A).$$
In this formulation, it may not be entirely obvious why we could have nontrivial cohomology operations
$$\def\H{{\sf H}}\H^m(X,A)→\H^n(X,B).$$
Consider now rewriting $\H^m(X,A)$ using the Dold–Kan correspondence:
$$\def\U{{\sf U}}\H^m(X,A)=[\N\Z[X],A[m]]=[\Z[X],Γ A[m]]=[X,\U Γ A[m]],$$
where $\U$ denotes the forgetful functor from simplicial abelian groups to simplicial sets.
Of course, the term $\def\K{{\sf K}}\U Γ A[m]=\K(A,m)$ is the $m$th Eilenberg–MacLane space of $A$.

In addition to providing us with a very simple proof that singular cohomology is representable by Eilenberg–MacLane spaces (which can be seen as another application), we also see immediately that cohomology operations
$$\H^m(X,A)→\H^n(X,B)$$
correspond to homotopy classes of maps
$$\K(A,m)→\K(B,n),$$
which themselves can be computed as elements of $\H^n(\K(A,m),B)$.
Thus, the cohomology groups of Eilenberg–MacLane spaces compute cohomology operations.

More generally, the Dold–Kan correspondence is the principal mediating tool between homological algebra and (nonabelian) homotopical algebra.  Indeed, the former term is predominantly used when working with chain complexes, and the latter when working with simplicial objects.

For example, the monoidal Dold–Kan correspondence establishes a Quillen equivalence between the model categories of simplicial rings and differential graded algebras.

In the commutative case, things become more interesting.
In characteristic 0, we can prove that simplicial commutative algebras are Quillen equivalent to commutative differential graded algebras.

This fails in characteristic $p>0$, where instead we have a Quillen equivalence between simplicial $\def\E{{\sf E}}\E_\infty$-algebras and differential graded $\E_\infty$-algebras, which again is established using the (appropriately refined) Dold–Kan correspondence.