The different mean squares (MS) for error measure how much variation is attributable to the total error, lack-of-fit error, and pure error. The mean squares equal the sums of squares divided by their degrees of freedom.
The mean square error (MSE) is the variance around the fitted values. MSE = Final SSE / DFE.
Minitab uses the means squares to calculate the p-value for the lack-of-fit test. Usually, you interpret the p-value instead of the mean squares.