Let $Q$ be an acyclic quiver. In here, we read that
there exists a bijection between $Q_0$ (set of vertices of $Q$) and the set $\{1,\dots,n\}$ such that if we have an arrow $i \to j$, then $j > i$; indeed such a bijection is constructed as follows. Let $1$ be any sink in $Q$, then consider the full subquiver $Q(1)$ of $Q$ having as set of points $Q_0 \setminus \{1\}$; let $2$ be a sink of $Q(1)$, and continue by induction. Such a numbering of the points of $Q$ is called an admissible numbering.
Let us do an example. Let $Q$ be the (acyclic) quiver: \begin{equation*} 1 \gets 2 \to 3 \gets 4 \to 5 \end{equation*} For example, $1$ is a sink, so we remove $1$ and we're left with: \begin{equation*} 2 \to 3 \gets 4 \to 5 \end{equation*} Then we decide to remove $5$, and so we get: \begin{equation*} 2 \to 3 \gets 4 \end{equation*} Now, only 3 is a sink, so we remove $3$ and I guess we're left with an empty quiver(?). So, as for our admissible numbering of $Q$, we got that the first 3 numbers are $\{1,5,3\}$ but what about the remaining $2$? Should our admissible numbering be $\{1,5,3,2,4\}$ or $\{1,5,3,4,2\}$? Or it doesn't matter? Can someone help me clarify what is the procedure in situations like this?
If you remove the vertex $3$ you are not left with the empty quiver but with the quiver with two vertices, $2$ and $4$. Since there are no arrows in the quiver you get two possible numbering which are admissible $\{1,5,3,2,4\}$ and $\{1,5,3,4,2\}$. Notice that you also made a choice beforehand, your first vertex could be either $1$, $3$, or $5$. In the next step you can remove another one of those three. In the third step it even depends on what you have done before, i.e. if your first two choices were $1$ and $3$, you can take $2$ as your third choice. In this theory it often only matters that there exists an admissible ordering and in the concrete constructions you work with a particular fixed one of those. But there will in general be many possible admissible numberings and not all the constructions are independent of the ordering.