$$0 \to A \xrightarrow{f}  B \xrightarrow{g} C \to 0$$

which is called an almost split sequence if
1) It does not split 
2) $f$ is a left almost minimal split and $g$ is a right almost minimal split (a lot is here packed, please see the definitions in AR paper, III)
3) $A$ and $C$ are indecomposable in $rep_k Q$ (category of finite-dimensional representations of quiver Q)

Such AR sequences are the main interest of Auslander-Reiten theory.  There is an AR sequence that can be easily observed from the AR quiver, which is

$$ 0 \to \tau V \to E \to V \to 0$$

where $V$ is some indecomposable and non-projective and $\tau V$ is the AR translate of $V$, which is the main subject of this post. There does not exist an AR translation of a projective representation since we begin with a projective resolution of an indecomposable, and if that is a projective, then it is not very interesting, and we fail to get a minimal projective resolution.

A projective resolution of a representation is helpful because of the equivalence between projectives and injectives using the Nakayama functor (which we may discuss in another post, but you may see this). The general idea is that  $$\nu : proj_k Q \to inj_k Q$$ and similarly, there exists an inverse $\nu^{-1}$. So, for any indecomposable $V$, we write its projective resolution as $$\cdots P_2 \to P_1 \to P_0 \to V \to 0$$ where $P_0 \to V$ is a surjection given that our category has enough projectives and, of course, our category is hereditary. We then have the AR translation of $V$ given by (for a minimal projective resolution)

$$0 \to \tau V  \to \nu P_1 \to \nu P_0$$  where $\nu$ is the Nakayama functor $\nu = DHom_A(-,A)$ which sends a projective to its corresponding injective.

For example, in $A_3$, if we take $V=110$ (where the digits are the dimension vectors of the vector spaces of representation $k \to k \to 0$), then we have the minimal projective resolution as $$ 001 \to 111 \to 110\to 0$$ and the AR translation of $110$ is $$0 \to \tau V = 011 \to 111 \to 100$$ (where $\tau V = ker (\nu (P_1) \to \nu(P_0))$ which, if you observe in the AR quiver, then it is the indecomposable which you reach after one step moving to the left side (if you can). Similarly, we can find the AR translation of all the other non-projective indecomposable. Using $\tau V$, we define the AR sequence $0 \to \tau V \to E \to V \to 0$ which is an almost split sequence (and dually, $0 \to V \to E \to \tau^{-1} V\to 0$). From here, we land into the discussion of extensions (derived functors!) and how to define these extensions using the AR quiver for which Auslander-Reiten wrote their beautiful formula (some other time).

If you could tell, there is a lot of homological algebra (and category theory) that exists between these steps, and Auslander-Reiten theory has interesting consequences in triangulated settings as well (see Happel).