Let's consider a few terms of the sequence $y$:
$ y[0] = \frac{x[0]+x[-1]+x[-2]}{3} $
$ y[1] = \frac{x[1]+x[0]+x[-1]}{3} $
$ y[2] = \frac{x[2]+x[1]+x[0]}{3} $
Notice how the values of $y$ are always an average of three values. Also, notice how the indices of $x$ "shift" to the right in the expressions, and the next value gets shifted in. This is as if we have a longer sequence, $\{x[-2],x[-1],x[0],x[1],x[2]\}$, and we have a window of three consecutive values of the sequence, and the window shifts over one place to the right for each term in the $y$ sequence. $\{\color{red}{x[-2],x[-1],x[0]},x[1],x[2]\}$ $\{x[-2],\color{red}{x[-1],x[0],x[1]},x[2]\}$ $\{x[-2],x[-1],\color{red}{x[0],x[1],x[2]}\}$
This is sometimes called a sliding-window average as well because of this property.
As for a concrete example, let's consider the sequence $ x[n] = n^2/10, \;\; n\geq 0 \\ x[n] = 0, \;\; n<0 $ This is plotted below. ![x[n]](https://i.stack.imgur.com/HxhBD.png)
Now, if you do the calculations, you average the first three points, then the second three points, then the third three points, etc. I leave the calculations out, but the result it as follows: 
The corresponding numerical values are given in the table below.
n x y -2 0 0 -1 0 0.0333 0 0 0.1667 1 0.1 0.4667 2 0.4 0.9667 3 0.9 1.6667 4 1.6 2.5667 5 2.5 3.6667 6 3.6 4.9667 7 4.9 6.4667 8 6.4 8.1667 9 8.1 10.0667 10 10 12.1667