Quiver Mechanics What do you suggest as an essential and introductory set of references in Physics literature for learning quivers? Any textbook?
 A: 
Besides, I like to know what kind of physical system quivers
represent,

Roughly speaking, quivers describe systems of D-branes. We have two important facts:

*

*It is known that D-branes on a given target space $X$ can be described by means of derived categories [1] [2]. When mathematicians talk about "derived categories" they try to understand the  derived bounded category of coherent sheaves over $X$, denoted by $\mathcal{D}^b(X)$; we can think physically on this object as all the possible possible D-brane bound states, up to the action of the renormalization group flow [3] [4]. More precisely, Any object in $\mathcal{D}^b(X)$ can be understood as a bound state of branes and anti-branes subject to the equivalence relation that declares that two ensembles are equivalent if we can perform "tachyon condensation" to go from one to the other [5].

*It is also known that D-brane bound states give a microscopic understanding of the black hole entropy within string theory.

By combining the two facts from above, we can conclude that derived categories are relevant to approach the problem of the exact computation the black hole degeneracies of a given black hole.
The key insight is that there are special classes of D-brane systems that can be associated to quivers (that's the achievement of the seminal paper "D-branes, Quivers, and ALE Instantons") and under nice conditions we can guarantee that the following is true:
$$D^b(\mathsf{Coh}(X))\simeq D^b(\mathsf{Rep}(Q)).$$
Then we conclude that when is possible to associate a quiver $Q$ to a D-brane system (such as black holes are) we can understand all its microstates by means of quiver representations.

I am interested in the map between the number of microstates in a specific configuration in SUGRA (e.g. multi-centered black holes) and quivers.

This is a question that can only be answered case by case. The most common one is the case of an extremal black hole in $\mathcal{N}=2, d=4$ supergravity. The chapter 2 of the Tudor Dan Dimofte PhD thesis is an excellent overview of the basic ideas.
My advice is try to understand the basics of the following (very pedagogical!) references; the first one explicity work out how to understand the higgs and coulomb branches of some simple quivers:
Quantum Quivers and Hall/Hole Halos

Split States, Entropy Enigmas, Holes and Halos
A: Try Quiver Representations and Quiver Varieties by Alexander Kirillov Jnr.
There are not many textbooks on "Quivers" since it is a highly specialized topic in String Theory and SUSY field theories. Also, see this answer and links therein to papers on the subject by Douglas and Witten.
