Earlier I said that the projection of the state vector onto one of the perpendicular axes of its Hilbert Space shows the contribution of that axes' eigenstate to the whole state. You may wonder what is meant by the whole state. You would think (and rightly so, according to classical physics) that our spin-1/2 particle could only exist entirely in one of the possible +1/2 and -1/2 states, and accordingly that its state vector could only exist lying completely along one of its coordinate axes. If the particle's axes are called x and y, and the state vector's x coordinate which denotes the contribution to the -1/2, or 0 state, and y coordinate which denotes the contribution to the +1/2, or 1 state, should only be (1,0), or (0,1).
That seems perfectly reasonable, but it simply is not correct. According to quantum physics a quantum system can exist in a mix of all of its allowed states simultaneously. This is the Principle of Superposition, and it is key to the power of the quantum computer. While the physics of superposition is not simple at all, mathematically it is not difficult to characterize this kind of behavior.