Linear combination of DT unit impulse
The plot that shows the sequence $x[n]$ is also called as the lollipop plot and it shows the individual samples of $x[n]$ with their weights and locations. 

Each lollipop represents a discrete-time unit-impulse signal, $\delta[n]$, which is located at the position it's shown and weighted by the value of the signal there. The superposition sum of all those impulses compose the signal $x[n]$.

Now, your signal has 3 lollipops located at $n = \{-1,0,1\}$ and weighted by $\{1,2,3\}$ respectively.

Since an impulse located at the position $n=d$, and weighted by $a$, is given by
$$ a \cdot \delta[n-d]$$

by using this, you can compose the signal $x[n]$ from its lollipops as:

$$ x[n] = 1 \cdot \delta[n-(-1)] + 2 \cdot \delta[n-(0)] + 3 \cdot \delta[n-(1)] $$  

$$ x[n] = 1 \cdot \delta[n+1] + 2 \cdot \delta[n] + 3 \cdot \delta[n-1] $$

Note that a generalization of this also yields the sifting-sum decomposition of any signal $x[n]$ as

$$x[n] = \sum_{k=-\infty}^{\infty} x[k] \delta[n-k] $$