The mean square error of an estimator $\hat{\theta}$ of a parameter $\theta$ in a statistical model is defined as: $$\operatorname{MSE}(\hat{\theta})\colon=\operatorname{E}\big[(\hat{\theta}-\theta)^2\big].$$

From the definition of the variance $\operatorname{Var}[X]=\operatorname{E}[X^2]-\operatorname{E}[X]^2$ , we can express the mean square error in terms of the bias by expanding the right hand side above: $$\operatorname{MSE}(\hat{\theta})=\operatorname{Var}\big[\hat{\theta}\big]+ \operatorname{Bias}(\hat{\theta})^2.$$

If $\hat{\theta}$ is an unbiased estimator, then its mean square error is identical to its variance: $\operatorname{MSE}(\hat{\theta})=\operatorname{Var}[\hat{\theta}]$ . An unbiased estimator such that $\operatorname{MSE}(\hat{\theta})$ is a minimum value among all unbiased estimators for $\theta$ is called a minimum variance unbiased estimator, abbreviated MVUE, or uniformly minimum variance unbiased estimator, abbreviated UMVU estimator.

Example. Suppose $X_1,X_2,\ldots,X_n$ are iid random variables ($n$ independent measurements of the radius of a coin, etc...) from a normal distribution $N(\mu,\sigma^2)$ (for example, $\mu$ would be the true radius of the coin, and $\sigma^2$ would be the error component of the measurements). Suppose $\overline{X}$ ($=\overline{X}_n$ ) is the sample mean. Then $\overline{X}$ is an unbiased estimator, so that $$\operatorname{MSE}(\overline{X})=\operatorname{Var}\left[\overline{X}\right]= \operatorname{Var}\left[\frac{1}{n}\sum_{i=1}^n X_i\right]=\frac{1}{n^2}\left(\sum_{i=1}^n \sigma^2\right)=\frac{\sigma^2}{n}.$$

Remark. The square root of MSE is called the ``root mean square error'', or rms error for short.