Lie groupoids in practice I am familiar with the notion of Lie groupoids. 
But, only easy examples of Lie groupoids I am familiar with are the following:


*

*Lie groupoids coming from manifolds; that are of the form $(M\rightrightarrows M)$.

*Lie groupoids coming from groups; that are of the form $(G\rightrightarrows *)$.

*Lie groupoids coming from an action of Lie group on a manifold, say $M\times G\rightarrow M$; that are of the form $(M\times G\rightrightarrows M)$, also called as translation Lie groupoid.


To understand some structure over a Lie groupoid, I would first see their special cases in above examples. This gives some understanding of what the structure is in some special cases. There is a theorem by Ieke Moerdijk and D. A. Pronk that says that any proper étale Lie groupoid is locally a translation groupoid. 
Are there other simpler Lie groupoids that you use in the same line of above mentioned Lie groupoids that gives a better understanding of a setup on an arbitrary Lie groupoid?  
 A: Equivalence relations $R\rightrightarrows M$ where $R\subseteq M\times M$, $s=p_1$, $t=p_2$, are groupoids (in sets). To get Lie you need the projections to be manifold submersions. It happens for example when the equivalence relation comes from a (nonsingular) foliation on a manifold. 
Edit: I see on wikipedia there's also what they call the "pair groupoid", which is an example of equivalence relation not arising from a (positive dimensional) foliation. $p_1,p_2:M\times M\rightrightarrows M$.
A: You can restrict your example 3 to a submanifold $N\subseteq M$. The groupoid consists of $(g,m)\in G\times N$ such that $g\cdot m \in N$.
A: My answer somewhat expands on Qfwfq's above. Lie groupoids are a useful tool to reduce certain infinite-dimensional transformation groups to a finite-dimensional setting. Moreover, as put by Alan Weinstein, they are a means of unifying the description of "internal" and "external" symmetries. As a typical example, which happens to be of importance in physics, one may consider $G$-bundle automorphisms $\psi$ of a principal $G$-bundle $\pi:P\rightarrow M$ over $M$, with the structure group $G$ being a finite-dimensional Lie group (from now on, we denote the projections of all $G$-bundles associated to $\pi$ collectively by $\tilde{\pi}$). This means that $\psi$ is a diffeomorphism of $P$ which commutes with the right action of $G$ on $P$. Since the orbits of this action are precisely the fibers of $P$, this entails that $\psi$ maps fibers of $\pi$ onto fibers of $\pi$ and therefore there is a unique diffeomorphism $\psi_M$ of $M$ such that $\pi\circ\psi=\psi_M\circ\pi$ (hence $\psi$ is a bundle automorphism of $P$ in the broader sense, covering $\psi_M$). If $\psi_M=\mathrm{id}_M$, we say that $\psi$ is a strict $G$-bundle automorphism of $\pi$ or a gauge transformation of $\pi$. Therefore, one can define the following three infinite-dimensional transformation groups:


*

*$\mathrm{Aut}_G(\pi)=$ group of $G$-bundle automorphisms of $\pi$;

*$\mathrm{Aut}^0_G(\pi)=$ group of gauge transformations of $\pi$;

*$\mathrm{Diff}(M)=$ group of diffeomorphisms of $M$, which appear naturally when considering any Lie groupoid over $M$ since by definition (see below) smooth bisections of Lie groupoids over $M$ yield elements of $\mathrm{Diff}(M)$ after composition with the latter's target maps. Particularly, $\mathrm{Diff}(M)$ is (up to a group isomorphism) the space of smooth bisections of the pair groupoid of $M$.
The first two act not only on $\pi$ but also on any $G$-bundle associated to $\pi$. It turns out that $\mathrm{Aut}^0_G(\pi)$ is a normal subgroup of $\mathrm{Aut}_G(\pi)$ and $\mathrm{Aut}_G(\pi)/\mathrm{Aut}^0_G(\pi)\cong\mathrm{Diff}(M)$ (the isomorphism being the map $[\psi]\rightarrow\psi_M$, where $[\psi]$ is the equivalence class of $\psi\in\mathrm{Aut}_G(\pi)$ modulo $\mathrm{Aut}^0_G(\pi)$), but $\mathrm{Diff}(M)$ is not generally (isomorphic to) a normal subgroup of $\mathrm{Aut}_G(\pi)$. A(n essentially) finite-dimensional treatment of $\mathrm{Aut}_G(\pi)$ and $\mathrm{Aut}^0_G(\pi)$ is made possible through the following finite-dimensional Lie groupoids, similarly as we expressed $\mathrm{Diff}(M)$ above as space of smooth bisections of the pair groupoid $M\times M\rightrightarrows M$. The idea is that the action of the above infinite-dimensional transformation groups can be reduced for most purposes to the action of the corresponding (finite-dimensional) Lie groupoids.


*

*Take the orbit space $(P\times P)/G$ of $P\times P$ modulo the diagonal action of $G$ on $P\times P$ induced by the right action of $G$ on $P$: $$(q_1,q_2)\cong(q_1,q_2)g\doteq(q_1g,q_2g)\ ,\quad g\in G\ ,\,q_1,q_2\in P\ .$$ Since $\pi\circ\mathrm{pr}_j(q_1,q_2)=\pi\circ\mathrm{pr}_j(q_1g,q_2g)=\pi(q_j)$ for all $q_1,q_2\in P$, $g\in G$, $j=1,2$, we see that $\pi\circ\mathrm{pr}_1,\pi\circ\mathrm{pr}_2:P\times P\rightarrow M$ respectively induce unique source and target maps $\sigma,\tau:(P\times P)/G\rightarrow M$ given by $\sigma((q_1,q_2)G)=\pi(q_1)$, $\tau((q_1,q_2)G)=\pi(q_2)$. Moreover, the pair groupoid multiplication $(q_1,q_2)\cdot(q_2,q_3)=(q_1,q_3)$ on $P\times P$ induces a multiplication map $$\mu:(P\times P)/G\,{}_\tau\!\!\times_\sigma(P\times P)/G\doteq\{((q_1,q_2)G,(q_3,q_4)G)\in(P\times P)/G\ |\ \pi(q_2)=\pi(q_3)\}\rightarrow(P\times P)/G$$ given by $$\mu((q_1,q_2)G,(q_3,q_4)G)=(q_1,q_4)G$$ whenever $\tau((q_1,q_2)G)=\pi(q_2)=\sigma((q_3,q_4)G)=\pi(q_3)$. Finally the composition of the diagonal map from $P$ to $P\times P$ with the quotient map modulo the diagonal action of $G$ on $P\times P$ defines a unit map $$\iota:M\ni p\mapsto(q,q)G\in(P\times P)/G$$ (where $q$ is any fixed element of $\pi^{-1}(p)$) and the inversion map $$\nu:(P\times P)/G\ni(q_1,q_2)G\mapsto (q_2,q_1)G\in(P\times P)/G$$ is then given in the obvious way. These five maps together define a Lie groupoid structure on $(P\times P)/G$, which is then called the gauge groupoid $\mathrm{Gau}_G(\pi)$ of $\pi$.

*The so-called strict or internal gauge groupoid $\mathrm{Gau}^0_G(\pi)$ of $\pi$ is simply the (associated $G$-)bundle of Lie groups $\tilde{\pi}:P\times_G G\rightarrow M$, with $G$ acting on itself by conjugation (this is sometimes called the adjoint bundle of $\pi$). Recall that $P\times_G G$ is the orbit space of $P\times G$ under the (right) $G$-action $$(q,h)g=(qg,g^{-1}hg)\ ,\quad q\in P\ ,\,g,h\in G\ .$$ This bundle embeds naturally into $(P\times P)/G$ through the map $$\epsilon:P\times G\ni(q,h)\mapsto\epsilon(q,h)\doteq(q,qh)\in P\times P\ ,$$ which is clearly seen to be compatible with the respective $G$-actions: $$\epsilon((q,h)g)=\epsilon(qg,g^{-1}hg)=(qg,qgg^{-1}hg)=(q,qh)g=\epsilon(q,h)g\ .$$ The image of the induced embedding $$\tilde{\epsilon}:P\times_G G\ni(q,h)G\mapsto\tilde{\epsilon}((q,h)G)\doteq(q,qh)G\ni(P\times P)/G$$ is simply the subset of $(P\times P)/G$ to which the restrictions of $\sigma$ and $\tau$ coincide - that is, the isotropy subgroupoid of $\mathrm{Gau}_G(\pi)$. As such, we can set the source and target maps $\sigma,\tau$ of $\mathrm{Gau}^0_G(\pi)$ as $$\sigma((q,h)G)=\tau((q,h)G)\doteq\pi(q)$$ and the multiplication map $\mu$ in $P\times_G G$ as $\mu((q_1,h_1)G,(q_2,h_2)G)\doteq(q_1,gh_1h_2)G$ whenever $\tilde{\pi}((q_1,h_1)G)=\pi(q_1)=\tilde{\pi}((q_2,h_2)G)=\pi(q_2)$, which amounts to $q_2=q_1g$ for a unique $g\in G$. Moreover, we can also set the unit map $$\iota:M\ni p\mapsto\iota(p)=(q,e)G\in P\times_G G\ ,$$ where $q$ is any element of $\pi^{-1}(p)$ and $e$ is the unit of $G$, and the inversion map $$\nu:P\times_G G\ni(q,h)G\mapsto\nu((q,h)G)=(q,h^{-1})G=(qh,e)G\in P\times_G G\ .$$ If we set $\mathrm{Gau}^0_G(\pi)$ as $P\times_G G$ endowed with such Lie groupoid operations, $\tilde{\epsilon}$ as defined above becomes a Lie groupoid (mono)morphism. More generally, a bundle of Lie groups over $M$ is always a Lie groupoid over $M$ for which the source and target maps coincide (with the bundle projection being, of course, any of those two), that is, a Lie groupoid which coincides with its own isotropy subgroupoid.
To make the connection of $\mathrm{Gau}_G(\pi)$ to $\mathrm{Aut}_G(\pi)$ and of $\mathrm{Gau}^0_G(\pi)$ to $\mathrm{Aut}^0_G(\pi)$, recall that a smooth bisection $\gamma:M\rightarrow G_1$ of a Lie groupoid $(\sigma,\tau):G_1\rightrightarrows M$ over $M$ is a smooth map such that $\sigma\circ\gamma=\mathrm{id}_M$ and $\tau\circ\gamma\in\mathrm{Diff}(M)$. If we denote by $\Gamma(G_1\rightrightarrows M)$ the space of smooth bisections of $G_1\rightrightarrows M$, we see that such a space is always a group if we set their product as follows: if $\gamma_1,\gamma_2\in\Gamma(G_1\rightrightarrows M)$, we write $(\gamma_1\gamma_2)(p)=\gamma_1\circ\tau\circ\gamma_2(p)$. The identity of $\Gamma(G_1\rightrightarrows M)$ is, of course, the unit map $\iota$, and the inverse of $\gamma\in\Gamma(G_1\rightrightarrows M)$ is given by $\gamma^{-1}=\nu\circ\gamma$, where $\nu$ is the inversion map. We can then establish the following group isomorphisms:


*

*$\mathrm{Aut}_G(\pi)\cong\Gamma(\mathrm{Gau}_G(\pi)\rightrightarrows M)$ through $\psi\mapsto(p\mapsto(q,\psi(q))G)$, where $q$ is any element of $\pi^{-1}(p)$. This yields $\sigma((q,\psi(q))G)=\pi(q)=p$ and $\tau((q,\psi(q))G)=\pi\circ\psi(q)=\psi_M\circ\pi(q)=\psi_M(p)$, as desired.

*$\mathrm{Aut}^0_G(\pi)\cong\Gamma(\mathrm{Gau}^0_G(\pi)\rightrightarrows M)$ through $\psi\mapsto(p\mapsto(q,g_q)G)$, where $q$ is any element of $\pi^{-1}(p)$ and $g_q\in G$ is the unique element such that $\psi(q)=qg_q$. Thus, $\sigma((q,g_q)G)=\tau((q,g_q)G)=\pi(q)=p$, as desired. We notice as well that $\Gamma(\mathrm{Gau}^0_G(\pi)\rightrightarrows M)=\Gamma(P\times_G G\rightarrow M)$.

*$\mathrm{Diff}(M)\cong\Gamma(M\times M\rightrightarrows M)$. We have already alluded to this isomorphism above, which is given by $\psi_M\mapsto(p\mapsto(p,\psi_M(p)))$.
In other words, we are in a sense "factoring out" the action of the structure group $G$ on the pair groupoid of $P$. This may restrict the elements of $\mathrm{Diff}(M)$ which appear after composition of smooth bisections with $\tau$ - for instance, if we set $\pi$ as the orthonormal frame bundle of the Riemannian manifold $(M,g)$ (where $g$ is some fixed Riemannian metric on $M$), the only diffeomorphisms of $M$ which appear in this way are the isometries of $(M,g)$.
