My understanding is that $\mathbf G_m$ stands for $k^*$ (the multiplicative group of the field $k$) as a group scheme.
But I have also seen symbols like $H^1(X_{\text{et}},\mathbf G_m)$. Is this talking about the first cohomology group of $X_{et}$ with coefficients in the constant sheaf with values in $\mathbf G_m$?
The first instance is the group scheme $\mathbb{G}_m := \mathrm{Spec}(k[x^{\pm 1}])$. Associated with this (abelian) group scheme there is its functor of points which induces a sheaf (for the étale topology in the OPs notation) of abelian groups. Note that if $X$ is a scheme over $k$, then the morphisms $X\to \mathrm{Spec}(k[x^{\pm 1}])$ are in canonical bijection with the $k$-algebra homomorphisms $k[x^{\pm 1}]\to\mathcal{O}_X(X)$; thus, with the set of units $\mathcal{O}_X(X)^\times$. Consequently, if we were talking about this sheaf on a fixed scheme $X$ with the Zariski topology, then the notation $\mathcal{O}^{\times}_{X}$ would be more common than $\mathbb{G}_m$. We use $\mathbb{G}_{m}$ to emphasise that it's the sheaf of groups for some site which may not be the usual Zariski topology.
