Do not understand formula... How does;
$x^{n} -y^{n}=(x-y)(x^{n-1} + x^{n-2}y+...+x y^{n-2}+ y^{n-1} )$
work on $x^{2} - y^{2}$ 
When I attempt to apply the formula on $x^{2} - y^{2}$ 
I get the following
$x^{2} - y^{2} =(x-y)( x^{1} + x^{0}y+...+x y^{0} + y^{1} )$
$x^{2} - y^{2} =(x-y)(2x+2y)$
which is obviously false. What is the correct way to use the formula?

You have too many terms, $x^{n-n}y^{n-1}$ is the last, which in this case is your second term $x^{n-2}y$. The third term would in your case be $x^{n-3}y^{n-2}$ which we don't want since $n=2$ and the exponent of $x$ would become negative.
To more easily see this, write the factorization in this form:
$$x^n - y^n = (x-y)(x^{n-1} + x^{n-2}y + \dots + x^{n-(n-1)}y^{n-2} + x^{n-n}y^{n-1})$$
In this case it becomes
$$x^n - y^n = (x-y)(x^1y^0 + x^0y^1) = (x-y)(x + y)$$
which of course is the correct factorization.
