A classification variable, or classification factor, is one which is assumed to have a qualitative effect upon the response, as opposed to a quantitative effect. The levels of a classification factor group the observations; the only modeling assumption is that the expected response is different within the different groups.
Classification variables are a subset of the factors. Thus, classification variables are listed both as factors and as classes. When you select factors, you are often asked which of those are classification variables. You can also use the 'Classes:' keyword to select classification variables. When you do, you are presented with a list of the current factors to choose from.
Note that factors with character (non-numeric) values are automatically designated classification variables.