Maybe this question will be clearer to me once I've read more about algebraic number theory but conversely maybe an answer to this question will help me at doing so.
Let $\mathcal{O}$ be a Dedekind domain, $L/\operatorname{Quot}(\mathcal{O})$ a finite extension and $\mathcal{O}_L$ the integral closure of $\mathcal{O}$ in $L$. Let further $\mathfrak{p} \subseteq \mathcal{O}$ be a prime ideal and $$ \mathfrak{p} \mathcal{O}_L = \mathfrak{P}_1^{e_1} \dots \mathfrak{P}_g^{e_g} $$ be its prime decomposition. Then, $\mathfrak{P}_i$ is called unramified if $e_i = 1$ and $\mathcal{O}_L/\mathfrak{P}_i$ is separable over $\mathcal{O}/\mathfrak{p}$.
I haven't yet really worked with this notion but it seems technical to require separability. So I want to ask why this is done.
This is answered quite satisfyingly in a blog entry by Alex Youcis about unramified morphisms.