You also have the rather new field of Leavitt Path Algebras (in which I happen to be working right now), where you take a field $K$ and a directed graph $E$, generate its extended graph $E'$ (add to $E$ its own edges reversed, denoted as $e^*$ for every edge $e$), and compute the Leavitt path algebra of $E$, $L(E)$, as the path algebra $KE'$ modulo some relations called the Cuntz-Krieger relations, inherited from the $C^*$-algebras setting, concretely:
(CK1) $e^* f=\delta_{ef}$ for any two edges $e,f$ of $E'$.
(CK2) $\sum_{e\in s^{-1}(v)}ee^* = v$, for $v$ a vertex which emits a nonzero finite number of edges, and $s^{-1}(v)$ the set of those edges.
(One can look at (CK1) and (CK2) as an abstract generalization of the product of matrix units).
These associative algebras provide us simultaneously with a purely algebraic analog of $C^*$-algebras of graph and a generalization of the Leavitt algebras (some associative algebras which do not satisfy the IBN property).
The full matrix rings over $K$ of order $n$ then arise as the Leavitt path algebras of the graphs with $n$ (consecutive) vertices and $n-1$ arrows, one between every pair of consecutive vertices.
Another simple example of Leavitt path algebra is the ring of Laurent polynomials over $K$, $K[x,x^{-1}]$, which appears associated to the graph with one vertex and a single loop.
The theory of LPAs is useful, and even beautiful, because:
They provide simple, visually attractive representations of well-known algebras.
They allow us to look at their algebraic properties by means of the combinatorial properties of their associated graphs. This happens to equip us with some rather powerful tools.
Conversely, they also enable "algebraic engineering", since they give us a straightforward, visual way to construct new algebras, customized with any algebraic or ring-theoretic properties we may desire. For example, we can show an algebra generated by five elements such that it is exchange but not purely innitely simple, by constructing a particular (small) graph with some (easy) graph-theoretic features.
Some references:
G. Abrams, G. Aranda Pino. "The Leavitt path algebra of a graph", J. Algebra 293 (2), 319-334 (2005). (Available at http://agt.cie.uma.es/~gonzalo/papers/AA1_Web.pdf).
P. Ara, M.A. Moreno, E. Pardo. "Nonstable K-Theory for graph algebras", Algebra Repr. Th. DOI 10.1007/s10468-006-9044-z (electronic). (Available at http://www.springerlink.com/content/pu701474q5300m63/).
G. Abrams, G. Aranda Pino, F. Perera, M. Siles Molina. "Chain conditions for Leavitt path algebras". (Available at http://agt.cie.uma.es/~gonzalo/papers/AAPS1_Web.pdf).
K.R. Goodearl. "Leavitt path algebras and direct limits", Contemp. Math. 480 (2009), 165-187.