Qubit, one or two complex numbers?

In the representation $|\psi\rangle = a|0\rangle + b|1\rangle $ we must have $|a|^2+|b|^2=1$, so that gives us one constraint. The second is that an overall global phase doesn't make any difference. We can use these two freedoms to chose $a$ and $b$ in a specific way. Traditionally we choose them such that $$|\psi\rangle = \cos \theta|0\rangle+\exp(i\phi)\sin \theta|1\rangle $$

See this wiki article on the Bloch sphere for details.