As long as $f$ is monic of degree $n$, and irreducible mod $p$, the $\mathbb{Z}/p^k$-algebra $(\mathbb{Z}/p^k)[x]/f(x)$ is flat and has perfect mod $p$ reduction $\mathbb{F}_{p^n}$. The theory of Witt vectors tells you that there is a unique such flat $\mathbb{Z}/p^k$-algebra, which can be identified with $W_{k}(\mathbb{F}_{p^n})=W(\mathbb{F}_{p^n})/p^k$. There is a Galois-theoretic way of thinking about $W(\mathbb{F}_{p^n})$: It is the unique unramified degree $n$-extension of the $p$-adic integers $\mathbb{Z}_p$.

The point is that the ring we get depends only on the mod $p$ reduction of your original polynomial in quite a canonical way.
As soon as you care about $\mathbb{Z}/p^k$-algebras which are not flat, or whose mod $p$ reduction is not perfect, this breaks down and things become messier.

EDIT Regarding units: For any $\mathbb{Z}/p^k$-algebra $A$, we have an exact sequence
$$
1\to (1+pA)\to A^{\times} \to (A/p)^{\times} \to 1,
$$
which is split if $A/p$ is perfect (there is a well-defined map $(A/p)^{\times} \to A^{\times}$, called Teichmüller lift).

In the flat case, where $A= W_k(A/p)$, one can further use the $p$-adic logarithm to identify the multiplicative subgroup $(1+pA)$ with the additive group $W_{k-1}(A/p)$ if $p$ is odd, or with $\{\pm 1\} \times W_{k-2}(A/2)$ is $p=2$. Thus,
$$
W_k(A/p)^{\times} \cong (A/p)^{\times} \times W_{k-1}(A/p) \text{ if $p$ odd},
$$
$$
W_k(A/2)^{\times} \cong (A/2)^{\times}\times \{\pm 1\} \times W_{k-2}(A/2) \text{ if $p=2$},
$$