a state vector
The state vector is the most complete description quantum mechanics offers of a system: a single vector, written |psi>, in Hilbert space, from which every prediction the theory can make is extracted. Where classical physics needs a list of positions and momenta, quantum physics packs everything knowable into this one abstract arrow.
It obeys the superposition principle — any linear combination of allowed states is itself an allowed state — and it is defined up to normalization and an overall phase, so physically a state is really a ray, the whole family c|psi> for any nonzero complex c. Concrete descriptions you have met are components of this same vector in different bases: the position-space wavefunction is psi(x) = <x|psi>, the momentum-space wavefunction is <p|psi>, the spin amplitudes are <up|psi> and <down|psi>. The vector is primary; those are its shadows.
One honest limitation: a single state vector describes a pure state — a system about which you have maximal quantum information. When a system is entangled with an environment or you have only statistical knowledge, a single vector no longer suffices and you must use a density matrix (a mixed state). Also, only relative phases within a superposition matter; the global phase of |psi> is unobservable, which is why the physical state is a ray rather than a literal vector.
A spin-1/2 state |psi> = a|up> + b|down> with |a|^2 + |b|^2 = 1 is a single vector in C^2; measuring spin along z gives up with probability |a|^2 and down with probability |b|^2.
Two complex numbers, one normalization, one ignorable global phase: the full state of a qubit.
A single state vector captures a pure state only; entanglement with an environment or mere ignorance forces you to a density matrix, and the global phase carries no physics.