Let $Q=(Q_0,Q_1)$ be a quiver with vertices $Q_0$ and arrows $Q_1$, A representation of a quiver, associates a vector space to each vertex. Let $\mathbb{V}=\oplus_{j\in Q_0} V_j$. Define the dimension vector as: $\mathbf{n}=(n_j)$, where $j\in Q_0$ and $n_j=\text{dim}(V_j)$ is the dimension of the $j$the vector space.
In some physical examples, it turns out that BPS states might be realised as the representations of a quiver with the dimension vector being mapped on to the charge vector of the BPS state. Below are three some references where quivers are used to describe halfBPS states in $\mathcal{N}=2$ quantum field theory and string theory.
 BPS Quivers and Spectra of Complete N=2 Quantum Field Theories
 The spectrum of BPS branes on a noncompact CalabiYau (the appendix has a nice introduction to quivers)
 Dbranes, Exceptional Sheaves and Quivers on CalabiYau manifolds: From Mukai to McKay

As discussed in reference 1, there are two kinds of walls across which the quiver description gets modified. As one crosses a wall, the quiver changes and in some (all?) cases, it can be understood as a mutation of a quiver. Additional data in the form of the central charge, a complex valued function, $Z(\mathbf{n})$ determines the walls. Mutations also provide a realization of Seiberg duality (see ref 4 above as well), In quivers with loops, there might be an associated superpotential (see references above for the definition of a superpotential).
A connection to Lie algebra follows from Gabriel's theorem and its generalizations which relates quivers to finite Lie algebras. One suspects that more general Lie algebras might be associated to all quivers but there is no such general theorem that I am aware of. However, see this paper: On the algebras of BPS states.