I am trying to understand about Lie groupoids but not able to get feeling for what it actually is.


So, question here is, 



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> What are Lie groupoids? How similar are they to Lie groups, Groupoids and what can one expect to do on a Lie groupoid?
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Any reference is appreciated.



What are Lie groupoids intuitively?
Here is an expansion of my comment, by request. The 2-category of Lie groupoids $\mathrm{LieGpd}$ admits the category of manifolds $\mathrm{Mfld}$ as a full sub-2-category (i.e. $\mathrm{Mfld} \to \mathrm{LieGpd}$ induces an iso on hom-groupoids), but the category of Lie groups $\mathrm{LieGrp}$ only has a functor $\mathrm{LieGrp} \to \mathrm{LieGpd}$, not full on 2-arrows, sending a group to the coresponding one-object groupoid. I think this is one reason why Lie groupoids are not best seen as a generalisation of Lie groups.


Given a group $G$ acting on $M$, there is a groupoid $M//G$ with objects $M$, morphisms $M\times G$ with source and target given by projection and action, respectively. This includes the boring case where $M=\*$, but the mix of potentially positive dimensional orbit space $M/G$, interesting orbits $mG$ nontrivial stabilisers $G\_m$ is more typical of what is happening in the Lie groupoid case. Nice Lie groupoids, for instance proper Lie groupoids $X$, where $(s,t)\colon X\_1\to X\_0\times X\_0$ is a proper map, locally look like group actions (times a codiscrete groupoid $U\times U \rightrightarrows U$), so this is sometimes a lot of what you need to consider anyway. Functors are just equivariant maps, where you can be equivariant with respect to a homomorphism $G\to H$.


Generalising from this to general Lie groupoids you need to forget the fact that the source map has identical fibres, and that there isn't some global group containing all the stabilisers.


