State vector is the vector representation of a quantum state.
Quantum state is often written as a linear combination of basis states where the coefficients are probability amplitudes. For example, a qubit is often written in the following way using Dirac notation.
$$\lvert\psi\rangle = a\lvert 0\rangle + b\lvert 1\rangle$$
But it's also a postulate of quantum mechanics that every quantum state lives in a Hilbert space. For qubits, that Hilbert space is $\mathbb{C}^2$. Hilbert space is by definition a complete inner product space. An inner product space is a vector space equipped with an inner product. Since $\mathbb{C}^2$ is a vector space, and $\lvert\psi\rangle$ is postulated to live in it, we can write the qubit as a column vector in $\mathbb{C}^2$:
$$\lvert\psi\rangle = \begin{pmatrix}a\\b\end{pmatrix}\qquad a,b\in\mathbb{C}$$
Because Hilbert space satisfies vector space axioms we can also do a linear decomposition:
$$\lvert\psi\rangle = a\begin{pmatrix}1\\0\end{pmatrix} + b\begin{pmatrix}0\\1\end{pmatrix}$$
From this, we see a natural definition of $\lvert 0\rangle$ and $\lvert 1\rangle$
$$\lvert 0\rangle = \begin{pmatrix}1\\ 0\end{pmatrix}$$ $$\lvert 1\rangle = \begin{pmatrix}0\\ 1\end{pmatrix}$$
But this is just a convention. We could have chosen any other complex numbers to do define $\lvert 0\rangle$ and $\lvert 1\rangle$ and quantum mechanics would have worked the same way. Thus a state vector is just a linear algebra representation of a quantum state.