Discriminant Analysis works by combining variables in such a way that the differences between predefined groups are maximized. The group membership must be known before using Discriminant Analysis.

Consider a simple two-group example. The aim is to combine (weight) the variable scores in some way so that a single new composite variable, the discriminant score, is produced. One way of thinking about this is in terms of a recipe, changing the proportions (weights) of the ingredients will change the characteristics of the finished food. At the end of the process it is hoped that each group will have a normal distribution of 'discriminant scores'; the degree of overlap between the discriminant score distributions can then be used as a measure of the success of the technique.
For example:

The discriminant scores are calculated from a Discriminant Function which has the form:
D = w1Z1 + w2Z2 + w3Z3 +.... wiZi

Where
D = discriminant score;
w i = weighting (coefficient) for variable i;
Z i = standardized score for variable i (a standardized variable has a mean of 0 and a standard deviation of 1).

Clearly the two groups can be separated by these two variables, but there is a large amount of overlap on each single axis (although the y variable is the 'better' discriminator). It is possible to construct a new axis which passes through the two group centroids ('means'), such that the groups do not overlap on the new axis.