There are several concepts often called a path category.
There is a forgetful functor from small strict categories to quivers. This forgetful functor has a left adjoint, giving the free category or path category of a quiver, whose objects are the vertices of the quiver. The morphisms from $a$ to $b$ in this free category are not merely the arrows from $a$ to $b$ in the quiver but instead are lists of the form $(a_n,f_n,a_{n-1},\ldots,a_{2},f_1,a_0)$ where $n \geq 0$ is a natural number, $a_0,a_1,\ldots,a_n$ are vertices of the graph, $a = a_0$, $b = a_n$, and for all $0 \lt i \leq n$, $f_i\colon a_{i-1} \to a_i$ is an edge from $a_{i-1}$ to $a_i$. The composition is given by the concatenation
whenever $a_0 = b_m$, and the target and source maps are given by $s(a_n,f_n,a_{n-1},\ldots,a_{2},f_1,a_0)=a_0$ and $t(a_n,f_n,a_{n-1},\ldots,a_{2},f_1,a_0) = a_n$. One informally writes $f$ for the morphism $(b,f,a)\colon a \to b$ in the free category and the identities of the free category are $id_a = (a,a)$; thus $f_n \circ f_{n-1} \circ \ldots \circ f_1 = (t(f_n),f_n,t(f_{n-1}),\ldots,t(f_1),f_1,s(f_1))$. The standard reference is Gabriel–Zisman.
Given a topological space $X$ (or some other kind of space with a notion of maps from intervals into it), there are various ways to obtain a small strict category whose objects are the points of $X$ and whose morphisms are paths in $X$. This is also often called a path category.
In particular, for every topological space there is its fundamental groupoid whose morphisms are homotopy classes of paths in $X$.
If $X$ is a directed space there is a notion of path category called the fundamental category of $X$.
When $X$ is a smooth space, there is a notion of path category where less than homotopy of paths is divided out: just thin homotopy. This yields a notion of path groupoid.
If parameterized paths are used, there is a way to get a category of paths without dividing out any equivalence relation: this is the Moore path category.
Given a category $X$, the functor category $[I,X]$ for $I$ the interval category might be called a “directed path category of $X$” (similar to path space). However, this functor category is referred to instead as the arrow category of $X$ and sometimes even denoted $Arr(X)$.