(Warning: I'm not an expert in the topic) Let's work in a "geometric" category, for example the category $\mathfrak{Diff}$ of "manifolds" (without the requirements of connectedness and second countability and even of finite dimensionality), and consider also the subcategory of "usual" finite dimensional (Edit: possibly disconnected, Hausdorff, second countable) manifolds $\mathrm{Diff}$. We can than consider two kinds of objects:
- Lie groupoids $\mathcal{G}=(\mathcal{G}_1 \rightrightarrows \mathcal{G}_0)$ where $\mathcal{G}_0$ and $\mathcal{G}_1$ are smooth manifolds (in $\mathfrak{Diff}$) and the source $\mathrm{src}$ and target $\mathrm{trg}$ maps are submersions.
- Pseudogroups $\Gamma$ of diffeomorphisms of some manifold $M$ in $\mathrm{Diff}$.
Remark 1. From a pseudogroup $\Gamma$ on $M$ (for example the local symplectomorphisms if $M$ is equipped with a symplectic form, or the local diffeomorphisms preserving a foliation...) an (abstract) groupoid $\mathcal{G}=\mathrm{Germ(\Gamma)}$ can be obtained via the construction of germs, and I believe it's also a (usually infinite dimensional) Lie groupoid. In this case $\mathcal{G}_0$ is some space of germs of manifolds, and $\mathcal{G}_1$ consists of germs of local diffeomorphisms belonging to $\Gamma$. (Edit: for the benefit of the readers let me say D.Carchedi in his answer below has pointed out that this $\mathcal{G}$ is actually finite dimensional).
Remark 2. Given a Lie group action $\varphi\colon K\times M \to M$, the "action groupoid" $G_\varphi=(K\times M \rightrightarrows M)$ in $\mathrm{Diff}$ can be constructed. If I'm not mistaken, also an "action pseudogroup" $\Gamma_\varphi$ can be constructed, which consists of all the restrictions of the diffeomorphisms of the form $\varphi_g$, $g\in K$, to open subsets of $M$.
Remark 3. We can apply the germ construction to $\Gamma_{\varphi}$, obtaining a new groupoid $\mathcal{G}_{\varphi} = \mathrm{Germ} (\Gamma_{\varphi})$. Given $\gamma\in \mathcal{G}$ and $x,y \in M$ such that $\mathrm{src} (\gamma)=\mathrm{germ}_M (x)$ and $\mathrm{trg} (\gamma)=\mathrm{germ}_M (y)$, we can think of the element $\gamma$ both as a $\mapsto$ arrow between points (with no internal structure) $x,y$ (since $\varphi_g \colon x\mapsto y$) and as a $\to$ arrow between objects $x$ and $y$ (once we identify $x$ and $y$ with their germs of neighbourhoods in $M$).
Remark 4. Groupoids can be viewed as a generalization of group actions (and, more generally, of equivalence relations). In the theory of orbifolds (either in the differentiable category or in others) the morphisms $G_1$ of a groupoid $G_1\rightrightarrows G_0$ in $\mathrm{Diff}$ representing the orbifold $X=[G_1\rightrightarrows G_0]$ are viewed as abstract "glueing data" to obtain a "space" $X$ by patching the pieces of the (possibly very disconnected) space $G_0$ (please correct me if this view is mistaken). So, ideally, a morphism $(g\colon x\to y) \in G_1$ corresponds to the assertion that $\varphi_g\colon x \mapsto y$ for some map $\varphi_g\colon U \to V$ between local patches around $x$ and $y$ (producing identifications in an "étale" way).
Remark 5. On the contrary to remark 4, in the theory of moduli (we leave for a moment the differentiable setting) when a groupoid $G_1\rightrightarrows G_0$ represents a moduli stack $\mathcal{M}$, elements (closed points) of $G_1$ are seen as maps (isomorphisms) between the "structured things" that $\mathcal{M}$ parametrizes.
Q1. Concerning Remark 3, which is the relation btween $G_{\varphi}$ and $\mathcal{G}_{\varphi}$? Are they equivalent?
Q2. Given a Lie groupoid $G=(G_1\rightrightarrows G_0)$, say with finite dimentional object space $M:=G_0$, which conditions (if any) on $G$ ensure that there is a pseudogroup $\Gamma$ on $M$ so that $G$ is equivalent to $\mathrm{Germ}(\Gamma)$ ? So, for example, can Remark 4 be made rigorous? In other words, can the $\mapsto$ viewpoint (examplified by Remark 4) be interchanged with the $\to$ viewpoint (examplified by Remark 5) ? Is étaleness of $G$ sufficient or necessary?
Q3. More generally, given a groupoid $\mathcal{G}=(\mathcal{G}_1 \rightrightarrows \mathcal{G}_0)$, is there a manifold $\mathcal{S}$ in $\mathfrak{Diff}$ such that $\mathcal{G}$ is equivalent to the groupoid $\mathrm{Germ} (\Gamma)$ of germs of some pseudogroup $\Gamma$ of diffeomerphisms of $\mathcal{S}$ ?
Q4. What about other "geometries" (i.e. other sites in which to consider internal groupoids and -assuming the notion can be generalized- pseudogroups)?
