Moments
Moments are measures of a distribution's shape and density.
Errors
Models generally assume that individual errors average to zero, i.e. the first moment of errors is zero: E[Ŷ - Y] = 0. Nonetheless, higher order moments are important.
The mean square error (MSE) is the second moment of the error: MSE(θ̂) = E[(θ̂ - E[θ̂])2]. MSE can be decomposed into the variance of the estimator and bias: MSE(θ̂) = Var(θ̂) + Bias(θ̂,θ)2 = Var(θ̂) + (E[θ̂]-θ)2.
Two important notes:
Bias, i.e. E[θ̂] - θ, is not the same as the first moment of errors.
If there is no bias, then MSE is the variance of the estimator: MSE(θ̂) = Var(θ̂).