When we have a sample of numbers, say
-69, 153, -54, 54, -198, -242, -63, 87, -45, -134, ...
we can calculate an estimation of the variance using the formula
$$\hat \sigma^2=s^2=\frac{\sum_{i=1}^n(x_i-\bar{x})^2}{n-1}$$
now this formula applies to any sequence of numbers, regardless of its underlying distribution. But what if we know that these numbers are sampled from a normal distribution, can we provide a better estimate of the variance? And/or if we know the true mean?
If the sample is known to be iid from a normal distribution with unknown mean and variance then $\hat{\sigma^2}$ is the Uniform Minimum Variance Estimator of $\sigma^2$.
If $\mu$ is known, then replace $\bar{x}$ by $\mu$ and delete the $-1$ in the denominator. That will be the UMVUE in that scenario.
If you don't know $\sigma^2$, but have correct knowledge about what it might be (a prior distribution), then the posterior mean of $\sigma^2$ will be biased but have smaller mean-squared error than the UMVUE.