This is pretty broad, but I'll give it a shot.

The origin (or at least one origin) of quivers in string theory is that, at a singularity, it is often the case that a D-brane becomes marginally stable against decay into a collection of branes that are pinned to the singularity. These are called "fractional branes". To describe the gauge theory that lives on the D-brane at the singularity, we get a gauge group for each fractional brane, and for the massless string states stretching between the D-brane, we get bifundamental matter. Thus, a quiver gauge theory.

The fractional branes and the bifundamental matter are essentially holomorphic information, so you can get at them by looking at the topological B-model. Since the B-model doesn't care about kaehler deformations, you can take a crepant resolution of the singularity which lets you deal with nice smooth things. The connection to the derived category of coherent sheaves comes about because the B-model (modulo some Hodge theoretic stuff) is essentially equivalent to the derived category (even though it doesn't matter so much any more, I can't resist plugging my paper, 0808.0168).

The equivalence of categories, in some ways, can be thought of as a tool for getting a handle on the derived category (representations are easier to deal with than sheaves) and the fractional branes, but I always thought there was some real physics there. Was never quite able to make those ideas work, though.

For the relation between representations and quiver reps, the easiest thing to say is that a representation of the quiver is the same as giving a vev to all bifundamentals.

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