Question: Hi,

The definition I have for a Path Algega of a quiver Q is that it is the algebra whose basis is formed by the oriented paths in Q, including the trivial ones. Apparently multiplication is given by concatenation of paths, and those that can't be concatenated are considered zero. That part I think I understand, but I am wondering if there is a non-formal interpretation of the addition operation on two elements of a path algebra someone could provide?

Answer: 
Maybe it is useful to add that a path algebra is a special case of a tensor algebra of a bimodule. Namely, the path algebra of a finite quiver $Q$ is the tensor algebra $T\_RV$ of a finite dimensional bimodule $V=V\_Q$ over a split semisimple finite dimensional commutative algebra $R=\oplus\_{i\in I}k$ over a field $k$. More precisely, if $M\_{ij}$ are the simple (1-dimensional) $R$-bimodules, then $V\_Q=\oplus n\_{ij}(Q)M\_{ij}$, where $n\_{ij}(Q)$ is the number of edges in $Q$ going from $i$ to $j$.
I realize that this question is four months old, but I'm new to the site.

Another thing you can do is use the relationship between path algebras and other algebras to breathe some life into the formal construction. For example, it is known that any elementary hereditary algebra over a field is isomorphic to a path algebra.