Quantum computing - Difference between a binary bit and a Q-bit
I think something worth pointing out early on is the ambiguity of the word state. In quantum information most things have very strong (shall I say natural?) physical interpretations / origins. 
In classical information we represent information as bits. A bit might correspond to the state of some classical system, such as an electric sign in a circuit, etc. Evolution of classical bit vectors is arbitrary in the sense that its enforced by us humans. 

In quantum computing information is represented using states of a quantum system (for example polarization of a photon). 
The states of a quantum system can be identified with vectors of a Hilbert space. Qubits $|0\rangle, |1\rangle$ are just an arbitrary choice bases of a two dimensional space. A two-qubit state represents a quantum system which is a composite of two two-dimensional subsystems. Evolution of quantum state vectors are governed by the laws of physics.

For a classical two-bit vector, the permissible values are $00,01,10,11$, whereas for a two-qubit system you can have any $a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle$ for $a,b,c,d\in\mathbb{C}$ provided that $|a|^2+|b|^2+|c|^2+|d|^2=1$. This, together with the postulates of quantum mechanics, permits a completely different range of behaviors of quantum systems (and qubit vectors), behaviors that are unseen in classical systems. This gives the above representation of information a lot more power compared to classical information representation. A number of physical phenomena allow various interesting information theoretic operations (such as teleportation, superdense coding, etc), which are not admissible using classical systems. 

The emphasis here really is on the fact that in quantum computing you can exploit a range of physical phenomena that is not typical in classical information theory.