Let's focus on the univariate case, where only one column contains missing data and we use all the other (completed) columns to impute the missing values before generalizing to a multivariate case.
mice actually has a few different imputation methods up its sleeve, each best suited for a particular use case. mice will often choose sensible defaults based on the data type (continuous, binary, non-binary categorical, and so on).
The most important method is what the package calls the norm method. This method is very much like stochastic regression. Each of the m imputations is created by adding a normal noise term to the output of a linear regression predicting the missing variable. What makes this slightly different than just stochastic regression repeated m times is that the norm method also integrates uncertainty about the regression coefficients used in the predictive linear model.
Recall that the regression coefficients in a linear regression...