Primitive characters are of no use in analyzing the relationship of organisms within a particular group. Were you to try to construct a family tree for all mammals, it is not helpful to note that they all have four legs. They do, but it doesn't help you in determining who is related to whom. In the jargon of cladistics, primitive characters are called plesiomorphic. Primitive characters shared by all members of the group in question are called symplesiomorphic.
Derived characters are advanced traits which only appear in some members of the group.
Cladistics is based on the assumption that the appearance of derived characters gives clues to evolutionary relationships. In our example, a derived character for some mammals might be loss of the tail, which occurs in the great apes and man. It is assumed that loss of the tail occurred only once, in the common ancestor of apes and man, and that none of us has one because we inherited that trait from our common ancestor. No animal within the group has a tail, yet our next-nearest relatives, the monkeys, do. It is important to note that the designation of primitive and derived characters has meaning only when related to the group under study.