What does 'OLS estimator for $\sigma^2$' mean? In the Wikipedia article on OLS (https://en.wikipedia.org/wiki/Ordinary_least_squares), it mentions that $S^2$ is the 'OLS estimator' for $\sigma^2$. I have no idea what this means. Does this notion of OLS have anything to do with OLS in the sense of having minimal error sum of squares?
 A: That is a long page.  Your quotation comes from about halfway down the page
This is the the result of trying to minimise the sum of squares of the residuals $y_i-\hat y _i$ from linear regression as you are presumably familiar with. The relevant definition of $s^2$ is higher up where it says

$s^2 = \frac{\hat\varepsilon ^\mathrm{T} \hat\varepsilon}{n-p} = \frac{(My)^\mathrm{T} My}{n-p} = \frac{y^\mathrm{T} M^\mathrm{T}My}{n-p}= \frac{y ^\mathrm{T} My}{n-p} = \frac{S(\hat\beta)}{n-p},\qquad
    \hat\sigma^2 = \frac{n-p}{n}\;s^2 $
The denominator, $n−p$, is the statistical degrees of freedom. The first quantity, $s^2$, is the OLS estimate for $σ^2$, whereas the second, $\hat\sigma^2$, is the MLE estimate for $\sigma^2$.

Rather than using matrices and vectors, you could say $s^2=\frac1{n-p}\sum\limits_{i=1}^n(y-\hat y_i)^2$ and if you were doing simple linear regression, i.e. with $p=1$ regressor or independent variable, you would have $s^2=\frac1{n-1}\sum\limits_{i=1}^n(y-\hat y_i)^2$, which may look more familiar.  This is the unbiased estimator of the variance of the error term $\varepsilon_i$ in your model.
A: To answer the question, consider the matrix formula for the Generalized Least-Squares case as indicated by Equation [4-15] in this educational reference, for example, to quote:
${Var[b | X] = σ^{2}(X’X)^{-1}}$
Now, if we consider the degenerate case of just one regression coefficient, the OLS variance estimate of this parameter (namely, the sample mean) becomes simply:
${Var[\mu | X] = σ{^2}/n}$
However, the suggested sample variance above is the uncorrected sample variance (where the correction factor is known as Bessel's correction).
As a supporting source, see discussion here, to quote:

The reason that an uncorrected sample variance, ${S^2}$, is biased stems from the fact that the sample mean is an ordinary least squares (OLS) estimator for μ: ${\overline {X}}$...

So, it does NOT actually have anything to do with OLS regression parameters themselves (in the degenerate case discussed, the unbiased sample mean, which produces the lowest value for the sum of squares) "in the sense of having minimal error sum of squares", as you asked.
