I want to say something like, "quivers (with relations) are finitely-generated unital algebras over $k^{\oplus n}$ ", but it's fairly vacuous, and I'm about to head off to the airport, so I don't have time to really think it through.
In physics, that is pretty much the situation once you throw in the word graded (it's easy to prove a version of Gabriel's theorem in this case). Quivers arise when you have a finite set of objects in a (pretriangulated dg-/A${}_\infty$/stable infinity/triangulated/whatever) category of D-branes, and the endomorphism algebra of the sum of these objects is presented as the path algebra of a quiver. If those objects form a nice generator, you get the usual equivalence between the original category and the derived category of quiver reps. The simple reps corresponding to the nodes of the quiver are called "fractional branes" in the physics literature, and the arrows in the quiver correspond to massless string states in the physics (as they are given by Ext^1s b/w the simple reps.)