As I understand it, if one has a complete knowledge of the state of a quantum system (insofar as one knows the statistical distributions of all the observables associated with the state) then one can represent it as a state vector (or ket) in an associated Hilbert space, and we say that the system is in a pure state. Furthermore, given an (orthonormal) basis for the Hilbert space, one can express this state vector as a linear combination of the basis vectors. This is usual carried out through the use of an eigenbasis induced by an operator acting on the Hilbert space, representing some observable. We say then that the state is in a quantum superposition.
What confuses me is how this differs from a mixed state? I understand that the situation is at least somewhat different since a mixed state arises when we have a lack of knowledge about the state (i.e. we lack all the possible information we could, in principle, have about it, that is, the statistical distributions of all the observables associated with the state). Hence, we must consider a statistical ensemble of possible pure states that the system could be in, each having an associated probability. This is a so-called classical probability as it does not stem from the intrinsic probabilistic nature of a quantum system, but rather the fact that we lack all the knowledge that we could possibly have about the system.
Is it simply, that in the case of a pure state, even though we know the statistical distribution of the observables associated with this state, we do not, a priori, before measuring the given observable, Know which eigenstate the quantum system is in, and hence must consider it to be in a quantum superposition of the available eigenstates? (In this case, it would be a so-called quantum probability, since such an uncertainty does not arise from lack of information about the state of the system, but is intrinsic in the quantum nature of the system).
Apologies for the long-windedness of this post, I just thought I'd write evergthing down on my thoguhts about it, and hopefully someone can correct/ explain it to me.
Edit: I think that perhaps my confusion stems from how to interpret a quantum superposition of states. How should one interpret this physically? (if I have an understanding of this perhaps it'll clear up things a little).