An observation is classified into a group if the squared distance (also called the Mahalanobis distance) of observation to the group center (mean) is the minimum. An assumption is made that covariance matrices are equal for all groups. There is a unique part of the squared distance formula for each group and that is called the linear discriminant function for that group. For any observation, the group with the smallest squared distance has the largest linear discriminant function and the observation is then classified into this group.
Linear discriminant analysis has the property of symmetric squared distance: the linear discriminant function of group i evaluated with the mean of group j is equal to the linear discriminant function of group j evaluated with the mean of group i.
This is for the simplest case, no prior probabilities or equal covariance matrices. If you consider Mahalanobis distance an adequate way to measure the distance of an observation to a group, then you do not need to make any assumptions about the underlying distribution of your data.
Minitab uses a single common covariance matrix to calculate the Mahalanobis distances between observations and classes. Also, Minitab calculates the linear discriminant functions (similar to regression coefficients), which can be used to classify new observations.
Use a linear analysis when you assume the covariance matrices are equal for all groups. Use a quadratic analysis when you assume the covariance matrices are not equal for all groups.