What is a moving average system? Can someone elaborate on what a moving average system is? 

I know that the system is defined as:
$$y[n] = \frac{x[n] + x[n-1] + x[n-2]}{3}$$
How would we draw $y[n]$ given that we have a graph with discrete values for $x[n]$? Can someone actually draw a sample discrete time $x[n]$ graph and show how the corresponding $y[n]$ graph is generated?
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
$$
x[n] = n^2/10, \;\; n\geq 0 \\
x[n] = 0, \;\; n&lt;0
$$
This is plotted below.


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
You need to convolve your data $\mathbf{x}$ with the impluse response of the corresponding FIR filter $\mathbf{h}$. You can learn the details from here.