To explain this let us call our state Psi. The states corresponding to unit vectors in H are in fact a special case of the density matrices. Now about superpositions versus mixtures. They teach you how to take superpositions in basic quantum mechanics. Given two unit vectors in H, one can take any linear combination of them and, if it's not zero, normalize it to be a unit vector again, which we call a superposition.
Mixtures are an utterly different sort of linear combination.
Phi This is called a mixture of Psi and Phi. Show that the states corresponding to unit vectors in Hilbert space are pure. Show that every density matrix is a mixture of states corresponding to unit vectors in Hilbert space. Show that such a state is pure if and only if its entropy is zero.