(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](http://en.wikipedia.org/wiki/Lie_groupoid) $\mathcal{G}=(\mathcal{G}\_1 \rightrightarrows \mathcal{G}\_0)$ 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](http://en.wikipedia.org/wiki/Pseudogroup) $\Gamma$ 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.




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> Q1. Concerning Remark 3, which is the relation btween $G\_{\varphi}$ and $\mathcal{G}\_{\varphi}$? Are they equivalent?
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> 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?
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> 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 $\mathcal{S}$ ?
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> Q4. What about other "geometries" (i.e. other sites in which to consider internal groupoids and -assuming the notion can be generalized- pseudogroups)?
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Groupoids vs Pseudogroups 
You've asked a mouthful. Let me do my best:


First, in your first remark, the Lie groupoid is NOT infinite dimensional. It is very finite dimensional. The arrows of $Germ(G)$ can be viewed as encoding a sheaf over $G\_0,$ and hence can be given the unique topology making the source map into a local homeomorphism. This endows the arrow space with the structure of a (possibly non-Hausdorff) manifold of the same dimension as $M$. If you do not like non-Hausdorff manifolds, you can instead construct a Lie $2$-groupoid out of purely Hausdorff manifolds which is equivalent to it.


Q1. Offhand, I am not familiar with the construction of a pseudo-group out of a group action, unless the group acting is discrete (but if you do, please give me a reference). Lets restrict to discrete $K$. In general, the action groupoid $G\_\varphi$ and the germ groupoid $\mathcal{G}\_\varphi$ are not equivalent, but there is a strong relation; the latter is the "effective-part" of the former. If $K$ is finite, they are the same if and only if the action if faithful. In general, it turns out that the former may be viewed as a gerbe over the latter (when considering their associated differentiable stacks)


Q2 and Q3: The condition you are looking for is that $G$ is an effective etale Lie groupoid. Etale means that the source and target maps are local diffeomorphisms. Effective is more subtle:


Given any arrow $g$ in an etale Lie groupoid $\mathcal{G}$, we can find a neighborhood $U$ of $g$ in the arrow space $\mathcal{G}\_1$ such that both the source and target maps restrict to an embedding over $U$. Hence $g$ induces a diffeomorphism from $s(U)$ to $t(U)$. This encodes a psuedo-group, and there is an obvious functor from $\mathcal{G}$ to the Lie groupoid associated to this pseudo-group. More or less by definition, an etale Lie groupoid is effective, if and only if this homomorphism is an isomorphism. Alternatively, effectivity means that for each point $x$ of $\mathcal{G}\_0$, the action of the stabilizer group of $x$ on the germ of $\mathcal{G}\_0$ around $x$ (given by the above construction) is faithful.


For your question on orbifolds, the answer is yes. See "Orbifolds, Sheaves, and Groupoids" by Moerdijk and Pronk.


Q4: If you make the question more specific, maybe I can attempt to answer it.


